If $A = <7 ,-5 ,6 >$ , $B = <4 ,-8 ,9 >$ and $C=A-B$ , what is the angle between A and C?

Question

1451 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-13T20:11:19

$theta ~~ 102.7°$

Compute vector C:

$barC=barA -barB = (7-4)hati + (-5 - -8)hatj + (6 - 9)hatk$

$barC = 3hati + 3hatj - 3hatk$

Compute the dot-product of vectors A and C:

$barA*barC = (7)(3) + (-5)(3) + (6)(-3)$

$barA*barC = -12$

There is another form for the equation of the dot-product that contains the angle between the two vectors:

$barA*barC = |barA||barC|cos(theta) = -12$

Compute the magnitude of vector A:

$|barA| = sqrt(7^2 + (-5)^2 + 6^2)$

$|barA| = sqrt(110)$

Compute the magnitude of vector C:

$|barC| = sqrt(3^2 + 3^2 + (-3)^2)$

$|barC| = 3sqrt(3)$

$sqrt(110)(3sqrt(3))cos(theta) = -12$

$cos(theta) = -12/(sqrt(110)(3sqrt(3))$

$theta = cos^-1(-4/(sqrt(330)))$

$theta ~~ 102.7°$

If A = <7 ,-5 ,6 >A = <7 ,-5 ,6 >, B = <4 ,-8 ,9 >B = <4 ,-8 ,9 > and C=A-BC=A-B, what is the angle between A and C?