What is the angle between $< 7, 2, - 3 >$ $<7 , 2 , -3 >$ and $< 8, 6, 0 >$ $< 8 , 6 , 0 >$ ?

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What is the angle between $< 7, 2, - 3 >$ $<7 , 2 , -3 >$ and $< 8, 6, 0 >$ $< 8 , 6 , 0 >$ ?

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Answer 1 · 2016-04-19T11:24:05

approximately $0.528 radians$ $0.528 " radians"$ or ${30.28}^{\circ}$ $30.28^@$

Explanation:

The angle, $θ$ $theta$ , between two vectors $u$ $u$ and $v$ $v$
is such that
$XXX cos (θ) = \frac{u \cdot v}{| | u | | \cdot | | v | |}$ $color(white)("XXX")cos(theta)=(u*v)/(abs(abs(u))*abs(abs(v)))$

For
$XXX u = < u_{1}, u_{2}, u_{3} > = < 7, 2, - 3 >$ $color(white)("XXX")u=< u_1,u_2,u_3 > = < 7,2,-3 >$
and
$XXX v = < v_{1}, v_{2}, v_{3} > = < 8, 6, 0 >$ $color(white)("XXX")v=< v_1,v_2,v_3 > = < 8,6,0 >$

$u \cdot v = u_{1} \cdot v_{1} + u_{2} \cdot v_{2} + u_{3} \cdot v_{3}$ $u*v= u_1*v_1+u_2*v_2+u_3*v_3$
$XXx = 7 \times 8 + 2 \times 6 + (- 3) \times 0 = 68$ $color(white)("XXx")=7xx8+2xx6+(-3)xx0=68$

$| | u | | = \sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}}$ $abs(abs(u)) = sqrt(u_1^2+u_2^2+u_3^2)$
$XXx = \sqrt{7^{2} + 2^{2} + {(- 3)}^{2}} = \sqrt{62}$ $color(white)("XXx")=sqrt(7^2+2^2+(-3)^2)=sqrt(62)$

$| | v | | = \sqrt{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}$ $abs(abs(v))=sqrt(v_1^2+v_2^2+v_3^2)$
$XXx = \sqrt{8^{2} + 6^{2} + 0^{2}} = \sqrt{100} = 10$ $color(white)("XXx")=sqrt(8^2+6^2+0^2)=sqrt(100)=10$

Therefore
$XXX cos (θ) = \frac{68}{10 \sqrt{62}}$ $color(white)("XXX")cos(theta)= 68/(10sqrt(62))$
$XXXXXXX \approx 0.863601$ $color(white)("XXXXXXX")~~0.863601$ (using a calculator)
and
$XXX θ = arccos (0.863601) \approx 0.528426$ $color(white)("XXX")theta=arccos(0.863601) ~~ 0.528426$ (radians)

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What is the angle between $< 7, 2, - 3 >$ $<7 , 2 , -3 >$ and $< 8, 6, 0 >$ $< 8 , 6 , 0 >$ ?

1 Answer

Explanation:

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What is the angle between <7,2,−3><7 , 2 , -3 > and <8,6,0> < 8 , 6 , 0 > ?

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What is the angle between $< 7, 2, - 3 >$ $<7 , 2 , -3 >$ and $< 8, 6, 0 >$ $< 8 , 6 , 0 >$ ?