If $A = < 3, 2, 5 >$ $A = <3 ,2 ,5 >$ , $B = < 5, - 7, 8 >$ $B = <5 ,-7 ,8 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?

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If $A = < 3, 2, 5 >$ $A = <3 ,2 ,5 >$ , $B = < 5, - 7, 8 >$ $B = <5 ,-7 ,8 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?

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Answer 1 · 2016-06-25T15:56:32

$θ = 87.12$ $theta=87.12$

Explanation:

$Strategy....$ $"Strategy...."$

$1 ⋄ find C = A - B$ $1 diamond" find " C=A-B$
$2 ⋄ find || A|| (magnitude of A)$ $2 diamond" find || A|| (magnitude of A)"$
$3 ⋄ find ||C|| (magnitude of C)$ $3 diamond" find ||C|| (magnitude of C)"$
$4 ⋄ find A.C (dot product of A and C)$ $4 diamond" find A.C (dot product of A and C)"$
$5 ⋄ use A.C=||A||.||C||. cos θ (dot product formula)$ $5 diamond" use A.C=||A||.||C||. cos" theta " (dot product formula)"$

$1 ⋄ ......................................................$ $1diamond" ......................................................"$

$A_{x} = 3; A_{y} = 2; A_{z} = 5$ $A_x=3" ; "A_y=2" ; "A_z=5$

$B_{x} = 5; B_{5} = - 7; B_{z} = 8$ $B_x=5" ; "B_5=-7" ; "B_z=8$

$C = < (A_{x} - B_{x}), (A_{y} - B_{y}), (A_{z} - B_{z}) >$ $C= < (A_x-B_x),(A_y-B_y),(A_z-B_z) >$

$C = < (3 - 5), (2 + 7), (5 - 8)$ $C= < (3-5),(2+7),(5-8)$

$we obtain :$ $"we obtain :"$
$C = < - 2, 9, - 3 >$ $C= < -2,9,-3>$

$2 ⋄ ........................................................$ $2diamond" ........................................................"$

$| | A | | = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}} = \sqrt{3^{2} + 2^{2} + 5^{2}} = \sqrt{38}$ $||A||=sqrt(A_x^2+A_y^2+A_z^2)=sqrt(3^2+2^2+5^2)=sqrt38$

$3 ⋄ ..........................................................$ $3diamond" .........................................................."$

$| | C | | = \sqrt{c_{x}^{2} + C_{y}^{2} + C_{z}^{2}} = \sqrt{{(- 2)}^{2} + 9^{2} + {(- 3)}^{2}}$ $||C||=sqrt(c_x^2+C_y^2+C_z^2)=sqrt((-2)^2+9^2+(-3)^2)$

$| | C | | = \sqrt{4 + 81 + 9} = \sqrt{94}$ $||C||=sqrt(4+81+9)=sqrt94$

$4 ⋄ ...........................................................$ $4diamond" ..........................................................."$

$A \cdot C = A_{x} \cdot C_{x} + A_{y} \cdot C_{y} + A_{z} \cdot C_{z}$ $A*C=A_x*C_x+A_y*C_y+A_z*C_z$

$A \cdot C = 3 \cdot (- 2) + 2 \cdot 9 + 5 \cdot (- 3) = - 6 + 18 - 15$ $A*C=3*(-2)+2*9+5*(-3)=-6+18-15$

$A \cdot C = - 3$ $A*C=-3$

$5 ⋄ ............................................................$ $5diamond" ............................................................"$

$- 3 = \sqrt{38} \cdot \sqrt{94} \cdot cos θ$ $-3=sqrt38*sqrt94*cos theta$

$cos θ = - \frac{3}{\sqrt{38 \cdot 94}}$ $cos theta=-3/(sqrt(38*94))$

$cos θ = - \frac{3}{59.766}$ $cos theta=-3/(59.766)$

$cos θ = 0.0501957635$ $cos theta=0.0501957635$

$θ = 87.12$ $theta=87.12$

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If $A = < 3, 2, 5 >$ $A = <3 ,2 ,5 >$ , $B = < 5, - 7, 8 >$ $B = <5 ,-7 ,8 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?

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Explanation:

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If A=<3,2,5>A = <3 ,2 ,5 >, B=<5,−7,8>B = <5 ,-7 ,8 > and C=A−BC=A-B, what is the angle between A and C?

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Explanation:

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If $A = < 3, 2, 5 >$ $A = <3 ,2 ,5 >$ , $B = < 5, - 7, 8 >$ $B = <5 ,-7 ,8 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?