Vector Operations

Key Questions

• How do force vectors affect an object in motion?

  Forces acting on any object change its movement.

  The change depends on the strength and directions of forces and on the way the object moved before forces acted upon it.

  To determine the result of forces acting on an object at any moment we have, first of all, add forces together according to the rules of adding the vectors (the rule of parallelogram). The resulting force is the source of acceleration of an object towards the direction of this force. The absolute value of the acceleration depends on the magnitude of the resulting force and the mass of an object.

  If, for example, the resulting force directed along the way the object is moving, it will continue moving along the same trajectory, but accelerating on its way.

  If the resulting force directed against the movement, the object will decelerate.

  If the resulting force acts at an angle to a trajectory of an object, it will deviate from that trajectory. The degree of deviation depends on the strength and direction of the resulting force and the mass of an object. The greater the mass - the less deviation from the trajectory would be observed.

  The best way to get deeper into this is to solve a few concrete problems with given movement and mass of an object and the forces acting upon it.

  Zor Shekhtman · 2 · Jul 10 2014

• What are vectors used for?
• Why vectors cannot be added algebraically?
• How do we represent the magnitude of a vector in physics?
• How do you find the equation of a vector orthogonal to a plane?
• Why are vectors important?
• How does a vector quantity differ from a scalar quantity?
• How can I calculate the magnitude of vectors?
• How do vectors subtract graphically?
• How do force vectors affect an object in motion?
• How can vectors be represented?
• How can vectors be combined?
• What are some examples of vectors?
• What are vectors?
• Question #5de41
• Question #8f0b0
• The two vectors A and B in the figure have equal magnitudes of 13.5 m and the angles are θ1 = 33° and θ2 = 110°. How to find (a) the x component and (b) the y component of their vector sum R , (c) the magnitude of R, and (d) the angle R ?
• What is the dot product of two vectors?
• Question #8599d
• Question #8bef5
• The wind exerts a force of 452 N north on a sailboat, while the water exerts a force of 325 N west on the sailboat. What is the magnitude and direction of the net force on the sailboat?
• Question #b00d1
• What are the parallelogram and the polygon methods?
• The #x# component of a vector is 8.7 units, and its #y# component is 06.5 units. What is the magnitude?
• What is the cross product of #[5, 6, -3]# and #[5, 2, 9]#?
• What is the cross product of #[1, -2, -3]# and #[3, 7, 9]#?
• What is the cross product of #[1, 3, 4]# and #[3, 7, 9]#?
• What is the cross product of #[1, -2, -3]# and #[1, 3, 4]#?
• What is the cross product of #[2, -5, 8] # and #[3, 7, 9]#?
• What is the cross product of #[1, -2, -3]# and #[2, -5, 8] #?
• What is the cross product of #[1, 3, 4]# and #[2, -5, 8] #?
• What is the cross product of #[1, 3, 4]# and #[3,2, 5] #?
• What is the cross product of #[3,2, 5]# and #[2, -5, 8] #?
• What is the cross product of #[3,2, 5]# and #[3, 1, 0] #?
• What is the cross product of #[2, 4, 5]# and #[2, -5, 8] #?
• What is the cross product of #[3,2, 5]# and #[2, 4, 8] #?
• What is the cross product of #[3,2, 5]# and #[1,2,-4] #?
• What is the cross product of #[3,2, 5]# and #[0,8,5] #?
• What is the cross product of #[0,8,5]# and #[2, 4, 8] #?
• What is the cross product of #[0,8,5]# and #[1,2,-4] #?
• What is the cross product of #[2, -1, 1]# and #[3,-6,4] #?
• What is the cross product of #[3, 0, 5]# and #[3,-6,4] #?
• What is the cross product of #[0,8,5]# and #[1, 4, -2] #?
• What is the cross product of #[1, 4, -2]# and #[3,-6,4] #?
• What is the cross product of #[1, 4, -2]# and #[2, -1, 1] #?
• What is the cross product of #[1, 4, -2]# and #[3, 0, 5] #?
• What is the cross product of #[3, 0, 5]# and #[2, -1, 1] #?
• What is the cross product of #[3, 0, 5]# and #[1,2,1] #?
• What is the cross product of #[1,2,1]# and #[2, -1, 1] #?
• What is the cross product of #[3,1,-5]# and #[2, -1, 1] #?
• What is the cross product of #[1,2,1]# and #[3,1,-5] #?
• What is the cross product of #[4,-3,2]# and #[3,1,-5] #?
• What is the cross product of #[4,-3,2]# and #[1,-2,3] #?
• What is the cross product of #[4,-3,2]# and #[-1,-1,2] #?
• What is the cross product of #[-1,-1,2]# and #[1,-2,3] #?
• What is the cross product of #[-3,0,1]# and #[1,-2,3] #?
• What is the cross product of #[-3,0,1]# and #[0,1,2] #?
• What is the cross product of #[2,4,5]# and #[0,1,2] #?
• What is the cross product of #[2,4,5]# and #[-1,0,1] #?
• What is the cross product of #[-1,0,1]# and #[0,1,2] #?
• What is the cross product of #[-1,0,1]# and #[3, 1, -1] #?
• What is the cross product of #[-3, 1, -1]# and #[0,1,2] #?
• What is the cross product of #[2,-1,2]# and #[5,1,-3] #?
• What is the cross product of #[2,-1,2]# and #[3,-1,2] #?
• What is the cross product of #[3,-1,2]# and #[5,1,-3] #?
• What is the cross product of #[2,-1,2]# and #[1,-1,3] #?
• What is the cross product of #[1,-1,3]# and #[5,1,-3] #?
• What is the cross product of #[2,-1,2]# and #[4,-3,1] #?
• What is the cross product of #[4,-3,1]# and #[1,-1,3] #?
• What is the cross product of #[3,-1,2]# and #[1,-1,3] #?
• What is the cross product of #[3,-1,2]# and #[-2,0,3] #?
• What is the cross product of #[-2,0,3]# and #[1,-1,3] #?
• What is the cross product of #[3,-1,2]# and #[1,-2,-1] #?
• What is the cross product of #[1,-2,-1]# and #[1,-1,3] #?
• What is the cross product of #[1,-2,-1]# and #[-2,0,3] #?
• What is the cross product of #[1,-2,-1]# and #[0, -1, 1] #?
• What is the cross product of #[1,-2,-1]# and #[4,3,6] #?
• What is the cross product of #[9,4,-1]# and #[4,3,6] #?
• What is the cross product of #[2, 1, -4]# and #[4,3,6] #?
• What is the cross product of #[9,4,-1]# and #[2, 1, -4] #?
• What is the cross product of #[1, -4, 0]# and #[4,3,6] #?
• What is the cross product of #<< -1, -1, 2 >># and #<< 4,3,6 >> #?
• What is the cross product of #[9,4,-1]# and #[-1, -1, 2] #?
• What is the cross product of #[-1, 2, 2]# and #[4,3,6] #?
• What is the cross product of #[3, 2, 5]# and #[4,3,6] #?
• What is the cross product of #[9,4,-1]# and #[2, 5, 4] #?
• What is the cross product of #[2, 5, 4]# and #[4,3,6] #?
• What is the cross product of #[2, 1, -4]# and #[1, -4, 0] #?
• What is the cross product of #[2, 1, -4]# and #[-1, -1, 2] #?
• What is the cross product of #[-1, -1, 2]# and #[1, -4, 0] #?
• What is the cross product of #[3, 2, 5]# and #[1, -4, 0] #?
• What is the cross product of #[2, 1, -4]# and #[3, 2, 5] #?
• What is the cross product of #[2, 5, 4]# and #[1, -4, 0] #?
• What is the cross product of #[-1, -1, 2]# and #[-1, 2, 2] #?
• What is the cross product of #[3, 2, 5]# and #[-1, 2, 2] #?
• What is the cross product of #[-1, -1, 2]# and #[3, 2, 5] #?
• What is the cross product of #[-1, -1, 2]# and #[2, 5, 4] #?
• What is the cross product of #[2, 5, 4]# and #[-1, 2, 2] #?
• What is the cross product of #[2, -1, 4]# and #[-1, 2, 2] #?
• What is the cross product of #[2, -1, 4]# and #[5, 2, -2] #?
• What is the cross product of #[4, -4, 4]# and #[5, 2, -2] #?
• What is the cross product of #[4, -4, 4]# and #[-6, 5, 1] #?
• What is the cross product of #[1, -3, 2]# and #[-8, 5, 1] #?
• What is the cross product of #[4, 0, 1]# and #[-1, 2, 3] #?
• What is the cross product of #[-1, 2, 3]# and #[-8, 5, 1] #?
• What is the cross product of #[-1, 2, 3]# and #[1, -3, 2] #?
• What is the cross product of #[-3, 5, -3]# and #[1, -3, 2] #?
• What is the cross product of #[-3, 5, -3]# and #[4, -11, 11] #?
• What is the cross product of #[3, 1, -4]# and #[1, 1, 18] #?
• What is the cross product of #[3, -4, 2]# and #[1, 1, 18] #?
• What is the cross product of #[3, 1, -4]# and #[3, -4, 2] #?
• What is the cross product of #[3, 1, -4]# and #[2, 6, -1] #?
• What is the cross product of #[2, 6, -1]# and #[1, 1, 18] #?
• What is the cross product of #[2, 6, -1]# and #[3, -4, 2] #?
• What is the unit vector that is normal to the plane containing <2i+7j-2k> and <8i-2j+3k>?
• What is the unit vector that is normal to the plane containing 3i+7j-2k and 8i+2j+9k?
• How do you normalize 3i+7j-2k?
• How do you normalize 8i+2j+9k?
• How do you normalize <0,2,0>?
• What is the unit vector that is normal to the plane containing <0,2,0> and <-1,1,1>?
• What is the unit vector that is normal to the plane containing <1,1,1> and <2,0,-1>?
• How do you normalize <2,0,-1>?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #(2i+ j - 3k)#?
• How do you normalize #(- 3 i + j -k)#?
• How do you normalize #(- 3 i + 12j -k)#?
• How do you normalize #(- 3 i + 12j -5k)#?
• How do you normalize #(- 7 i -j -25k)#?
• How do you normalize #(- 7 i -j +25k)#?
• How do you normalize #(7 i -j +2k)#?
• How do you normalize #(2 i -3j -k)#?
• What is the unit vector that is normal to the plane containing #(2i - 3 j +k )# and #(2i+ j - 3k)#?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #(2i - 3 j +k )#?
• How do you normalize #(2i - 3 j +k)#?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #(- 2i - j - k)?
• How do you normalize #(- 2i - j - k)#?
• What is the unit vector that is normal to the plane containing #(3i + 4j - k)# and #(2i+ j - 3k)#?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #(3i + 4j - k)#?
• What is the unit vector that is normal to the plane containing #(- 4i + 5 j-k)# and #(2i+ j - 3k)?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #(- 4i + 5 j - 3k)?
• How do you normalize #(- 4i + 5 j- k)#?
• How do you normalize #(i+2j+2k)#?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #(i+2j+2k)?
• What is the unit vector that is normal to the plane containing #(i+2j+2k)# and #(2i+ j - 3k)?
• What is the unit vector that is normal to the plane containing #(- 3 i + j -k)# and #( i +k )?
• What is the unit vector that is normal to the plane containing #( i +k )# and #(2i+ j - 3k)?
• What is the unit vector that is normal to the plane containing #( i +k)# and #(i+2j+2k) #?
• How do you normalize #(i+k)#?
• How do you normalize #( i - 2 j + 3 k )#?
• What is the unit vector that is normal to the plane containing # ( i +k) # and # ( i - 2 j + 3 k ) #?
• What is the unit vector that is normal to the plane containing # ( i - 2 j + 3 k ) # and # (i+2j+2k) #?
• How do you normalize #( i + 7 j + 4 k )#?
• What is the unit vector that is normal to the plane containing # ( i +k) # and # ( i + 7 j + 4 k) #?
• What is the unit vector that is normal to the plane containing # ( i - 2 j + 3 k) # and # ( i + 7 j + 4 k) #?
• How do you normalize #( - 4 i - 5 j + 2 k )#?
• What is the unit vector that is orthogonal to the plane containing # ( - 4 i - 5 j + 2 k) # and # ( i + 7 j + 4 k) #?
• What is the unit vector that is orthogonal to the plane containing # ( i - 2 j + 3 k) # and # ( - 4 i - 5 j + 2 k) #?
• What is the unit vector that is orthogonal to the plane containing # ( i - 2 j + 3 k) # and # (4 i + 4 j + 2 k) #?
• What is the unit vector that is orthogonal to the plane containing # ( i - 2 j + 3 k) # and # ( - 5 i + 4 j - 5 k) #?
• What is the unit vector that is orthogonal to the plane containing # ( - 4 i - 5 j + 2 k) # and # (4 i + 4 j + 2 k) #?
• What is the unit vector that is orthogonal to the plane containing # ( - 5 i + 4 j - 5 k) # and # (4 i + 4 j + 2 k) #?
• What is the unit vector that is orthogonal to the plane containing # ( - 4 i - 5 j + 2 k) # and # (- 5 i + 4 j - 5 k) #?
• How do you normalize #( - 4 i - 5 j + 2 k )#?
• How do you normalize #( - 5 i + 4 j - 5 k )#?
• What is the projection of #<3, -6, 2># onto #<1, 1, 1>#?
• How do you normalize #<3, -6, 2>#?
• What is the unit vector that is orthogonal to the plane containing # <3, -6, 2> # and # <1, 1, 1> #?
• What is the projection of #<< 3, -6, 2 >># onto #<< 3, 2, 1 >>#?
• How do you normalize #<3, 2, 1>#?
• What is the unit vector that is orthogonal to the plane containing # <3, -6, 2> # and # <3, 2, 1> #?
• What is the unit vector that is orthogonal to the plane containing # <3, 2, 1> # and # <1, 1, 1> #?
• How do you normalize #<0, 1, 3>#?
• What is the projection of #<3, -6, 2># onto #<0, 1, 3>#?
• What is the projection of #<0, 1, 3># onto #<3, 2, 1>#?
• What is the unit vector that is orthogonal to the plane containing # <0, 4, 4> # and # <1, 1, 1> #?
• What is the unit vector that is orthogonal to the plane containing # <0, 4, 4> # and # <1, -1, 1> #?
• How do you normalize #<0, 4, 4>#?
• What is the projection of #<0, 1, 3># onto #<0, 4, 4>#?
• What is the unit vector that is orthogonal to the plane containing # (29i-35j-17k) # and # (41j+31k) #?
• What is the unit vector that is orthogonal to the plane containing # (32i-38j-12k) # and # (41j+31k) #?
• What is the unit vector that is orthogonal to the plane containing # (29i-35j-17k) # and # (32i-38j-12k) #?
• How do you normalize # (18i -30j -12k) #?
• What is the projection of #(32i-38j-12k)# onto # (18i -30j -12k)#?
• What is the unit vector that is orthogonal to the plane containing # (20j +31k) # and # (32i-38j-12k) #?
• What is the unit vector that is orthogonal to the plane containing # (29i-35j-17k) # and # (20j +31k) #?
• What is the unit vector that is orthogonal to the plane containing # (i + j - k) # and # ( i - j + k) #?
• What is the projection of #(-i + j + k)# onto # ( i - j + k)#?
• What is the unit vector that is orthogonal to the plane containing # (-i + j + k) # and # ( i - j + k) #?
• What is the projection of #(-i + j + k)# onto # ( i - j + k)#?
• What is the projection of #(3i + 2j - 3k)# onto # ( i - j + k)#?
• What is the projection of #(-i + j + k)# onto # ( 3i + 2j - 3k)#?
• What is the unit vector that is orthogonal to the plane containing # (3i + 2j - 3k) # and # ( i - j + k) #?
• What is the unit vector that is orthogonal to the plane containing # (-i + j + k) # and # (3i + 2j - 3k) #?
• What is the projection of #(-i + j + k)# onto # ( i -2j + 3k)#?
• What is the projection of #(i -2j + 3k)# onto # ( 3i + 2j - 3k)#?
• How do you normalize # (i -2j + 3k) #?
• What is the unit vector that is orthogonal to the plane containing # (i -2j + 3k) # and # ( i - j + k) #?
• What is the unit vector that is orthogonal to the plane containing # (-i + j + k) # and # (i -2j + 3k) #?
• What is the unit vector that is orthogonal to the plane containing # (3i + 2j - 3k) # and # (i -2j + 3k) #?
• How do you normalize # (2i+j+2k) #?
• What is the projection of #(i -2j + 3k)# onto # ( 2i+j+2k)#?
• What is the unit vector that is orthogonal to the plane containing # (3i + 2j - 3k) # and # (2i+j+2k) #?
• What is the projection of #(j+2k)# onto # ( i+2j)#?
• How do you normalize # (8i + 12j + 14k) #?
• What is the unit vector that is orthogonal to the plane containing # (8i + 12j + 14k) # and # (2i+j+2k) #?
• How do you normalize # (3i – 4j + 4k) #?
• What is the projection of #(8i + 12j + 14k)# onto # (3i – 4j + 4k)#?
• What is the unit vector that is orthogonal to the plane containing # (3i – 4j + 4k) # and # (2i+j+2k) #?
• What is the unit vector that is orthogonal to the plane containing # (8i + 12j + 14k) # and # (3i – 4j + 4k) #?
• How do you normalize # (2i + 3j – 7k) #?
• What is the projection of #(8i + 12j + 14k)# onto # (2i + 3j – 7k)#?
• What is the projection of #(2i + 3j – 7k)# onto # (3i – 4j + 4k)#?
• What is the unit vector that is orthogonal to the plane containing # (8i + 12j + 14k) # and # (2i + 3j – 7k) #?
• What is the unit vector that is orthogonal to the plane containing # (2i + 3j – 7k) # and # (3i – 4j + 4k) #?
• What is the unit vector that is orthogonal to the plane containing # (3i + 2j - 6k) # and # (3i – 4j + 4k) #?
• What is the unit vector that is orthogonal to the plane containing # (2i + 3j – 7k) # and # (3i + 2j - 6k) #?
• How do you normalize # (3i + 2j - 6k) #?
• What is the projection of #(3i + 2j - 6k)# onto # (3i – 4j + 4k)#?
• How do you normalize # (3i - j - 2k) #?
• What is the projection of #(3i - j - 2k)# onto # (3i – 4j + 4k)#?
• What is the projection of #(3i + 2j - 6k)# onto # (3i - j - 2k)#?
• What is the unit vector that is orthogonal to the plane containing # (3i - j - 2k) # and # (3i – 4j + 4k) #?
• What is the unit vector that is orthogonal to the plane containing # (2i + 3j – 7k) # and # (3i - j - 2k) #?
• How do you normalize # (-2i- 3j + 2k) #?
• What is the projection of #(3i + 2j - 6k)# onto # (-2i- 3j + 2k)#?
• What is the unit vector that is orthogonal to the plane containing # (-2i- 3j + 2k) # and # (3i – 4j + 4k) #?
• What is the unit vector that is orthogonal to the plane containing # (2i + 3j – 7k) # and # (-2i- 3j + 2k) #?
• How do you normalize # (5i- 3j + 12k) #?
• How do you normalize # (15i- 3j + 12k) #?
• What is the dot product of #<5,1,-7># and #<8,4,9>#?
• What is the dot product of #<-2,5,-7># and #<8,-3,1>#?
• What is the dot product of #<7,-3, 9 ># and #<4,2,-9 >#?
• What is the dot product of #<5,-3,6 ># and #<-8,2,-9 >#?
• What is the dot product of #<3,-7,6 ># and #<4,1,7 >#?
• What is the dot product of #<3,5,-3 ># and #<-3,6,3 >#?
• What is the dot product of #<1,5,-3 ># and #<-8,3,-5 >#?
• What is the dot product of #<1,5,-3 ># and #<-1,4,2 >#?
• What is the dot product of #<1,4,-7 ># and #<-1,6,2 >#?
• What is the dot product of #<-5,2,8 ># and #<-1,6,2 >#?
• What is the dot product of #<1,0,-2 ># and #<-1,-4,1 >#?
• What is the dot product of #<-4,-3,8 ># and #<3,-1,7 >#?
• What is the dot product of #<-7,2,-1 ># and #<0,8,-27 >#?
• What is the dot product of #<9,1,-1 ># and #<6,-8,2 >#?
• What is the dot product of #<3,4,1 ># and #<5,-1,2 >#?
• What is the dot product of #<7,1,1 ># and #<-2,4,-8>#?
• What is the dot product of #<7,0,1 ># and #<-2,4,-3>#?
• What is the dot product of #<-4,0,-2 ># and #<3,-9,1>#?
• What is the dot product of #<-4,5,-2># and #<-2,4,1>#?
• What is the dot product of #<1,-4,5 ># and #<-5,7,3 >#?
• What is the dot product of #<5,6,-3 ># and #<-4,8,-5 >#?
• What is the dot product of #<5,4,-4># and #<-4,1,5 >#?
• What is the dot product of #<8,-1,4># and #<9,1,5 >#?
• What is the dot product of #<8,4,1># and #<4,-2,3 >#?
• What is the dot product of #<8,8,1># and #<-1, 2,3 >#?
• What is the dot product of #<-1,-2,1># and #<-1, 2,3 >#?
• What is the dot product of #<7,-8,1># and #<4,-6,3 >#?
• What is the dot product of #<9,4,1># and #<7,6,3 >#?
• What is the dot product of #<9,4,1># and #<-5,1,7 >#?
• What is the dot product of #<2,4,7># and #<1,-7,7 >#?
• What is the dot product of #<2,0,7># and #<-4,9,-1 >#?
• What is the dot product of #<-1,4,7># and #<9,-7,5 > #?
• What is the dot product of #<-6,1,0># and #<2,7,5 > #?
• What is the dot product of #<-6,4,2 > # and #<7,1,0 > #?
• What is the dot product of #<-8,4,2 > # and #<-3,1,6 > #?
• What is the dot product of #<-8,5,2 > # and #<-3,1,7 > #?
• What is the dot product of #<-8,1,2 > # and #<-2,5,-1 > #?
• What is the dot product of #<-8,1,2 > # and #<-2,-8,-1 > #?
• What is the dot product of #<-1,1,2 > # and #<-2,-8,-1 > #?
• What is the projection of #<8,2,1 ># onto #<9,6,3 >#?
• What is the projection of #<-6,2,1 ># onto #<-5,1,3 >#?
• What is the projection of #<7,-5,6 ># onto #<-1,-3,7 >#?
• What is the projection of #<8,-1,6 ># onto #<-1,5,-3 >#?
• What is the projection of #<3,-4,6 ># onto #<-1,7,-3 >#?
• What is the projection of #<6,-4,8 ># onto #<-1,3,-7 >#?
• What is the projection of #<3,4,-1 ># onto #<-1,3,-6 >#?
• What is the projection of #<5,6,1 ># onto #<8,-5,1 >#?
• What is the projection of #<5,8,-2 ># onto #<4,-5,2 >#?
• What is the projection of #<5,0,-2 ># onto #<1,-1,0 >#?
• What is the projection of #<3,-7,4># onto #<1,4,0 >#?
• What is the projection of #<8,-5,3 ># onto #<7,6,0 >#?
• What is the projection of #<6,5,3 ># onto #<2,-1,8 >#?
• What is the projection of #<6,0,3 ># onto #<2,5,-7 >#?
• What is the projection of #<6,-4,9 ># onto #<7,8,7 >#?
• What is the projection of #<2,-5,1 ># onto #<6,-8,3 >#?
• What is the projection of #<2,-7,1 ># onto #<4,-5,9 >#?
• What is the projection of #<2,7,1 ># onto #<3,-5,7 >#?
• What is the projection of #<7,1,1 ># onto #<-1,6,-8 >#?
• What is the projection of #<4,9,1 ># onto #<-1,7,2 >#?
• What is the projection of #<4,0,1 ># onto #<6,-3,7 >#?
• What is the projection of #<-5,3,7 ># onto #<0,8,-2 >#?
• What is the projection of #<-8,3,7 ># onto #<1,4,-3 >#?
• What is the projection of #<-2,3,7 ># onto #<1,6,1 >#?
• What is the projection of #<-2,2,5 ># onto #<1,6,9 >#?
• What is the projection of #<-6,1,5 ># onto #<1,5,9 >#?
• What is the projection of #<4,-6,3 ># onto #<1,5,2 >#?
• What is the projection of #<2,-4,3 ># onto #<1,2,2 >#?
• What is the projection of #<2,4,3 ># onto #<-3,4,-1 >#?
• What is the projection of #<-4,8,9 ># onto #<8,1,-1 >#?
• What is the projection of #<7,-8,3 ># onto #<5,-6,1 >#?
• What is the projection of #<5,8,3 ># onto #<2,4,-2 >#?
• What is the projection of #<-2,3,3 ># onto #<2,-3,2 >#?
• What is the projection of #<-3,5,-9 ># onto #<2,-1,4 >#?
• What is the projection of #<8,2,-6 ># onto #<5,-1,7 >#?
• What is the projection of #<4,-1,6 ># onto #<1,3,-2 >#?
• What is the projection of #<-6,3,2 ># onto #<1,4,-3 >#?
• What is the projection of #<-2,4,2 ># onto #<1,8,-3 >#?
• What is the projection of #<5,-2,3 ># onto #<1,-6,-3 >#?
• What is the projection of #<-5,2,8># onto #<8,-3,3 >#?
• What is the projection of #<-5,2,8># onto #<4,-6,3 >#?
• What is the projection of #<4,-6,8># onto #<4,-6,3 >#?
• What is the norm or #<5,2,-7 >#?
• What is the norm of #<1,-3,-2 >#?
• What is the norm or #<1,6,-1 >#?
• What is the norm or #<-7 ,6,-1 >#?
• What is the norm or #<-7 , 5, 1 >#?
• What is the norm of #< 7 , 5, 1 >#?
• What is the norm or #< 7 , -3 , 2 >#?
• What is the norm or #< 4 , 3 , -2 >#?
• What is the norm or #< 4 , 3 , 9 >#?
• What is the norm or #< 4 , -4 , 9 >#?
• What is the norm or #< 8 ,-4 , 9 >#?
• What is the norm of #< 8 ,-2 , 5 >#?
• What is the norm of #< -5 , 4 , 5 >#?
• What is the norm of #< 5 , 7 , -2 >#?
• What is the norm of #< -3, -4 , -2 >#?
• What is the norm of #< -3, -1 , 8 >#?
• What is the norm of #< -3, -1 , -3 >#?
• What is the norm of #< 5 , -2, 3 >#?
• What is the norm of #< -5 , -2, 4 >#?
• What is the norm of #< -5 , -2, -1 >#?
• What is the norm of #< 7 , -2, -1 >#?
• What is #|| <-3, 6 , 8 > - <5, -2 , 7 > ||#?
• What is #|| <-3, -6 , 8 > - <5, -2 , -3 > ||#?
• What is #|| <-1, -2, 7 > - <3, 2 , -3 > ||#?
• What is #|| <-1, -2, 4 > - < 2, -6, 1 > ||#?
• What is #|| <-1, 7, 4 > - < 2, 1, 1 > ||#?
• What is #|| <-1, 9, -2 > - < 9 , -4, 1 > ||#?
• What is #|| <-1, -9, -2 > - < -2 , -4, 1 > ||#?
• What is #|| <-1, 4, 2 > - < -2 , -7, 3 > ||#?
• What is #|| <7, 3, -1> - < -2 , 5, 3 > ||#?
• What is #|| <-8, 3, -1> - < 4 , -5, 3 > ||#?
• What is #|| <-8, 5, 1> - < 4 , -7, 9 > ||#?
• What is #|| <-3, 5, 1> - < 4 , -7, 4 > ||#?
• What is #|| <-6, 1, -3 > - < 4 , 6, 5 > ||#?
• What is #|| <-7, 31, -3 > - < 4 , 4, -1 > ||#?
• What is #|| <-7, -7, -3 > - < 4 , 4, -1 > ||#?
• What is #|| <-9, 3, 3 > - < 6, -2, 1 > ||#?
• What is #|| <-9, 3, 3 > - < 9, -2, 1 > ||#?
• What is #|| <4, 5, -7 > - < 9, -2, 1 > ||#?
• What is #|| <4, 9, 7 > - < 9, -1, 2> ||#?
• What is #|| <1, 9, 7 > - < 2, -1, 2> ||#?
• What is #|| <1, 9, 7 > - < 4, -2, -9 >||#?
• What is #|| <1, 6, -3 > - < 4, -2, -9 >||#?
• What is # || < 5 , -6 , 9> + < 2 , -4, -7 > || #?
• What is # || < 5 , 3, -3 > || #?
• What is # || < -6 , 3, -3 > || #?
• What is # || < -6 , -7 , -3 > || #?
• What is #||<< -6,-7,1 >>||# (the magnitude of #<< -6, -7, 1 >>#)?
• What is # || < -2 , -7 , 1 > || #?
• What is # || < -2 , 3 , 1 > || #?
• What is # || < -2 , 3 , 6 > || #?
• What is # || < 9 , 1 , 6 > || #?
• What is # || < -4 , 1 , 6 > || #?
• What is # || < -4 , 8 , 6 > || #?
• What is # || < -4 , 8 , -5 > || #?
• What is # || < 4 , -3 , -8 > || #?
• What is # || < 4 , 9 , -8 > || #?
• What is # || < -3, 9 , 5 > || #?
• What is # || < -3, -3, 0 > || #?
• What is # || < -3, -3, 0 > || #?
• What is # || < 3, 4 , 0 > || #?
• What is the cross product of #<6,-2,8 ># and #<1,3,-4 >#?
• What is the cross product of #<9,2,8 ># and #<1,3,-4 >#?
• What is the cross product of #<9,2,8 ># and #<1,8,4 >#?
• What is the cross product of #<9,2,8 ># and #<6, -2, 7 >#?
• What is the cross product of #<-3,5,8 ># and #<6, -2, 7 >#?
• What is the cross product of #<-3,5,8 ># and #<6, -2, 7 >#?
• What is the cross product of #<-3,5,8 ># and #<-9,3,1>#?
• What is the cross product of #<-3,4, 8 ># and #<-9,3,1>#?
• What is the cross product of #<-3,4, 8 ># and #<2,-4, 1>#?
• What is the cross product of #<-3,4, 8 ># and #<2,-4, -9 >#?
• What is the cross product of #<-3, -1, 8 ># and #<2,-4, -9 >#?
• What is the cross product of #<-3, -1, 8 ># and #<-1, 4, -9 >#?
• What is the cross product of #<-3,0, 8 ># and #<-1, -2, 9 >#?
• What is the cross product of #<-3,0, 2># and #<-1, -2, 9 >#?
• What is the cross product of #<-3,0, 2># and #<-1, -4, 1>#?
• What is the cross product of #<8, 4 ,-2 ># and #<-1, -4, 1>#?
• What is the cross product of #<8, 4 ,-2 ># and #<-1, -1, 6 >#?
• What is the cross product of #<8, 4 ,-2 ># and #<-1, 3 ,2 >#?
• What is the cross product of #<-2, 5 ,-2 ># and #<-1, 3 ,2 >#?
• What is the cross product of #<-2, 5 ,-2 ># and #<-1, -9 ,2 >#?
• What is the cross product of #<-2, 5 ,-2 ># and #<7 , -9 ,2 >#?
• What is the cross product of #<-2, 5 ,-2 ># and #<7 ,3 ,2 >#?
• What is the cross product of #<-2, -1 ,-6 ># and #<7 ,3 ,2 >#?
• What is the cross product of #<3 ,1 ,-6 ># and #<7 ,3 ,2 >#?
• What is the cross product of #<3 ,1 ,-6 ># and #<-2 ,3 ,2 >#?
• What is the cross product of #<3 ,1 ,-6 ># and #<-2 ,-4 , 8 >#?
• What is the cross product of #<3 ,1 ,-6 ># and #<-2 ,1 , 8 >#?
• What is the cross product of #<3 ,-4 ,-3 ># and #<-2 ,1 , -7 >#?
• What is the cross product of #<-3 ,-6 ,-3 ># and #<-2 ,1 , -7 >#?
• What is the cross product of #<-3 ,-6 ,-3 ># and #<0 ,1 , -7 >#?
• What is the cross product of #<< -3,-6,-3 >># and #<< -5,2,-7 >>#?
• What is the cross product of #<7 ,4 ,-3 ># and #<-5 ,2 , -7 >#?
• What is the cross product of #<7 ,4 ,1 ># and #<-5 ,2 ,8 >#?
• What is the cross product of #<0 ,4 ,1 ># and #<-5 ,2 ,8 >#?
• What is the cross product of #<10 ,4 ,1 ># and #<-5 ,2 ,3 >#?
• What is the cross product of #<5, -3,1 ># and #<-5 ,2 ,3 >#?
• What is the cross product of #<5, -3,1 ># and #<-2 ,2 ,3 >#?
• What is the cross product of #<5, -3, 8 ># and #<-2 ,7 ,3 >#?
• What is the cross product of #<5, 2 , 8 ># and #<4 ,7 ,3 >#?
• What is the cross product of #<5, 2 ,5 ># and #<4 ,7 ,3 >#?
• What is the cross product of #<5, 2 ,5 ># and #<4 ,1 ,3 >#?
• What is the cross product of #<5, 2 ,5 ># and #<4 ,1 ,9 >#?
• What is the cross product of #<5, 2 ,15 ># and #<4 ,1 ,9 >#?
• What is the cross product of #<-1, 2 ,0 ># and #<-3 ,1 ,9 >#?
• What is the cross product of #<-1, 2 ,7 ># and #<-3 ,1 ,4 >#?
• What is the cross product of #<-1, 2 ,7 ># and #<-3 ,1 ,-1 >#?
• What is the cross product of #<-1, 2 ,27 ># and #<-3 ,1 ,-1 >#?
• What is the cross product of #<7, 2 ,5 ># and #<-3 ,1 ,-6 >#?
• What is the cross product of #<7, 2 ,6 ># and #<-3 ,5 ,-6 >#?
• What is the cross product of #<7, 5 ,6 ># and #<3 ,5 ,-6 >#?
• What is the cross product of #<7, 5 ,6 ># and #<3 ,5 ,-2 >#?
• What is the cross product of #<2 , 5 ,-3 ># and #<3 ,5 ,-2 >#?
• What is the cross product of #<2 , 5 ,-3 ># and #<5 ,6 ,-2 >#?
• What is the cross product of #<2 , 5 ,-7 ># and #<5 ,6 ,-9 >#?
• What is the cross product of #<5 , 5 ,-7 ># and #<5 ,6 ,-3 >#?
• What is the cross product of #<5 , 5 ,-7 ># and #<5 ,1 ,-3 >#?
• What is the cross product of #<4 , 5 ,-7 ># and #<5 ,1 ,-3 >#?
• What is the cross product of #<4 , 5 ,-9 ># and #<5 ,1 ,-3 >#?
• What is the cross product of #<4 , 5 ,-9 ># and #<5 ,-3 ,-3 >#?
• What is the cross product of #<4 , 5 ,-9 ># and #<4, 3 ,-3 >#?
• What is the cross product of #<4 , 5 ,-9 ># and #<4, 3 ,0 >#?
• What is the cross product of #<4 , 5 ,-9 ># and #<4, 8 ,-2 >#?
• What is the cross product of #<4 , 5 ,-9 ># and #<4, 1 ,-2 >#?
• What is the cross product of #<4 , 5 ,0 ># and #<4, 1 ,-2 >#?
• What is the component form of a vector with an initial point of (-2, 3) and a terminal point of (-4, 7)?
• The resultant of the two forces #3N# and#2N# at an angle #theta# is doubled when first force in increased to #6N#. Find #theta#?
• Question #aecf3
• Consider the angle sum formulas: #sin(A+B)=sinA cosB+cosA sinB# #cos(A+B)=cosA cosB−sinA sinB# show that #tan(90^0+θ)=− 1/tanθ#. and use this identity to prove that lines #y = mx and y = m_1x# are perpendicular iff #mm_1=−1#?
• Question #74c1d
• Question #ae8d8
• Question #848d3
• Question #89967
• How will you prove the trigonometric formula #cos(A+B)=cosAcosB-sinAsinB# by using formula of cross product of two vectors ?
• How will you prove the formula #sin(A+B)=sinAcosB+cosAsinB# using formula of scalar product of two vectors?
• How will you prove the formula #sin(A-B)=sinAcosB-cosAsinB# using formula of scalar product of two vectors?
• How will you prove the formula #cos(A-B)=cosAcosB+sinAsinB# using formula of vector product of two vectors?
• Question #be172
• Question #d7dc2
• Question #d3574
• What is the direction of null vector?
• What is the angle between two forces of equal magnitude, #F_a# and #F_b#, when the magnitude of their resultant is also equal to the magnitude of either of these forces?
• Question #c976d
• Question #8f4d7
• Question #e27ab
• Question #84ddc
• Determine unit vector which is perpendicular to both A=2i+j+k and B=i-j+2k?
• Question #4760f
• Question #c380e
• Question #15905
• Question #4bd8a
• Question #b8942
• Question #1b250
• Can a vector have a negative component?
• What is the value of #(A #x# B)^2# + #(A * B)^2# ?
• Question #4acab
• Question #e748d
• Question #82dbe
• Question #960ba
• Why is vector division not possible?
• Question #a16d3
• Question #fc58a
• Question #5d77a
• A farmer wants to wall off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure, where A = 4.99 km, B = 2.46 km, and C = 3.23 km and then correctly calculates the length and orientation of D?
• Question #2a277
• Why does multiplication of vectors not have a corresponding division?
• Question #1f88b
• Question #a42a8
• Question #2d7d4
• Question #89959
• Question #4dd1c
• Question #276ee
• Question #18491
• Question #987b7
• Question #5c77f
• Question #795a2
• Question #5b790
• Question #0b785
• If A= A1i + A2j + A3k and B= B1i + B2j + B3k. Prove that A.B= A1B1+A2B2+A3B3?
• Question #fca5d
• Question #17931
• Sarah can paddle a rowboat at 6 m/s in still water. She heads out across a 400 m river at an angle of 30 upstream. She reaches the other bank of the river 200 m downstream from the direct opposite point from where she started. Determine the river current?
• Question #50b52
• Question #26a9b
• If we assume that the definition of the inner product of the normalized vectors #hatu_1# and #hatu_2# is #2u_1^2 + u_2^2# (these are vector components), how do you use the Gram-Schmidt process to generate orthonormal vectors from #vecv_1# and #vecv_2#?
• Question #834e1