If $A = < 4, - 5, - 4 >$ $A = <4 ,-5 ,-4 >$ , $B = < - 9, 1, - 6 >$ $B = <-9 ,1 ,-6 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?

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If $A = < 4, - 5, - 4 >$ $A = <4 ,-5 ,-4 >$ , $B = < - 9, 1, - 6 >$ $B = <-9 ,1 ,-6 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?

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Answer 1 · 2016-02-26T12:10:32

$0.801 radians or {45.67}^{\circ}$ $0.801 " radians" or 45.67^@$

Explanation:

If $A = < 4, - 5, - 4 >$ $A=< color(blue)(4, -5, -4) >$
and $B = < - 9, 1, - 6 >$ $B = < color(green)( -9, 1, -6) >$
then
$XXX C = A - B = < 4 - (- 9), - 5 - 1, - 4 - (- 6) >$ $color(white)("XXX")C= A-B= < color(blue)4-(color(green)(-9)), color(blue)(-5)-color(green)(1), color(blue)(-4)-(color(green)(-6)) >$

$XXXX = < 13, - 6, 2 >$ $color(white)("XXXX")=< color(red)(13, -6, 2) >$

$θ_{A C} = arccos (\frac{A \cdot C}{| | A | | \times | | C | |})$ $theta_(AC) = arccos ((A*C)/(||A||xx||C||))$

$A \cdot C = 4 \times 13 + (- 5) \times (- 6) + (- 4) \times 2$ $A*C= color(blue)(4)xxcolor(red)(13) +color(blue)((-5))xxcolor(red)((-6))+color(blue)((-4))xxcolor(red)(2)$
$XXX = 76$ $color(white)("XXX")= 76$

$| | A | | = \sqrt{4^{2} + {(- 5)}^{2} + {(- 4)}^{2}}$ $||A|| = sqrt(color(blue)(4)^2+color(blue)((-5))^2+color(blue)((-4))^2)$
$XXX = \sqrt{57}$ $color(white)("XXX")=sqrt(57)$

$| | C | | = \sqrt{13^{2} + {(- 6)}^{2} + 2^{2}}$ $||C|| =sqrt(color(red)(13)^2+color(red)((-6))^2+color(red)(2)^2)$
$XXX = \sqrt{209}$ $color(white)("XXX")=sqrt(209)$

$θ = arccos (\frac{76}{\sqrt{57} \cdot \sqrt{209}}) = arccos (\frac{4}{\sqrt{33}})$ $theta= arccos(76/(sqrt(57)*sqrt(209))) = arccos(4/sqrt(33))$

$X = 0.801 radians or {45.67}^{\circ}$ $color(white)("X")=0.801 " radians" or 45.67^@$

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If $A = < 4, - 5, - 4 >$ $A = <4 ,-5 ,-4 >$ , $B = < - 9, 1, - 6 >$ $B = <-9 ,1 ,-6 >$ 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=<4,−5,−4>A = <4 ,-5 ,-4 >, B=<−9,1,−6>B = <-9 ,1 ,-6 > and C=A−BC=A-B, what is the angle between A and C?

1 Answer

Explanation:

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If $A = < 4, - 5, - 4 >$ $A = <4 ,-5 ,-4 >$ , $B = < - 9, 1, - 6 >$ $B = <-9 ,1 ,-6 >$ and $C = A - B$ $C=A-B$ , what is the angle between A and C?