What is the angle between #<-3,9,-7 > # and #< 4,-2,8 >#?

1 Answer
Feb 20, 2016

#theta~= 2.49 # radians

Explanation:

Note: The angel between two nonzero vector u and v, where #0 <= theta <= pi# is define as

#vec u = < u_1, u_2,u_3 > #

#vec v = < v_1, v_2,v_3 > #

#cos theta= (u *v )/(||u|| " ||v|| #

Where as: #" " u *v= (u_1v_1) + (u_2v_2) + (u_3v_3)#

#||u|| = sqrt((u_1)^2 +(u_2)^2+(u_3)^2)#

#||v||=sqrt((v_1)^2 +(v_2)^2+(v_3)^2)#

Step 1: Let

#vec u = < -3, 9, -7 > # and
#vec v= < 4, -2, 8 > #

Step 2: Let's find #color(red)(u *v)#

#color(red)(u *v) = (-3)(4) + (9)(-2) + (-7)(8)#

#= -12 -18 -56#
#= color(red)(-86)#

Step 3: Let find #color(blue)(||u||)#

#vec u= < -3, 9 - 7>#

#color(blue)(||u||) = sqrt((-3)^2 + (9)^2 + (-7)^2)#

#=sqrt(9+81+49)#

#=color(blue)(sqrt139)#

Step 4 Let find #color(purple)(||v||)#

#vec v = < 4, -2, 8>#

#color(purple)(||v||) = sqrt((4)^2 + (-2)^2 + (8)^2)#

#= sqrt(16 + 4 + 64) =color(purple)(sqrt84)#

Step 5; Let substitute it back to the formula given above, and find #theta#

#cos theta= (u *v)/(||u|| " ||v||) #

#cos theta = color(red)(-86)/((color(blue)sqrt(139))color(purple)((sqrt84))#

#cos theta = color(red)(-86)/(sqrt11676)#

#theta= cos^(-1)(-86/(sqrt11676))#

#theta~= 2.49# radians

**note: this is because #u *v <0#