How do you use the definition of the scalar product, find the angles between the following pairs of vectors: A = 3i + 4j - 4k and B= 4i - 5j + 5k?

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Answer 1 · 2016-12-15T19:18:03

Please see the explanation.

The scalar product as two definitions:

$"[1] "barA*barB = (a_x)(b_x) + (a_y)(b_y) + (a_z)(b_z)$
$"[2] "barA*barB = ||barA||||barB||cos(theta)$

where $theta$ is the angle between the vectors.

Use [1] to compute the scalar product:

$barA*barB = (3)(4) + (4)(-5) + (-4)(5)$

$barA*barB = -28$

Compute the magnitude of $barA and barB$

$||barA|| = sqrt(3^2 + 4^2 + (-4)^2)$

$||barA|| = sqrt(41)$

$||barB|| = sqrt(4^2 + (-5)^2 + 5^2)$

$||barB|| = sqrt(66)$

Use [2] to find the value of $theta$

$"[2] "barA*barB = ||barA||||barB||cos(theta)$

$-28= sqrt(41)sqrt(66)cos(theta)$

$-28/(sqrt(41)sqrt(66)) = cos(theta)$

$theta = cos^-1(-28/(sqrt(41)sqrt(66)))$

$theta ~~ 122.565^@$