What is the angle between $<1 , 3 , -4 >$ and $< 5 , -2 , 2 >$ ?

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Answer 1 · 2016-10-02T17:18:16

$theta ~~ 1.833 radians$

Let $bar A = <1, 3, -4>$
Let $bar B = <5, -2, 2>$

$bar A•bar B = (1)(5) + (3)(-2) + (-4)(2)$

$bar A•bar B = -9$

$|barA| = sqrt(1² + 3² + (-4)²)$

$|barA| = sqrt26$

$|barB| = sqrt(5²+ (-2)² + 2²)$

$|barB| = sqrt33$

$bar A•bar B = |barA||barB|cos(theta)$ where $theta$ is the angle between the vectors

Substitute the known values and then solve for $theta$ :

$-9 = sqrt26sqrt33cos(theta)$

$theta = cos^-1(-9/(sqrt26sqrt33))$

$theta ~~ 1.833 radians$

What is the angle between <1 , 3 , -4 > <1 , 3 , -4 > and < 5 , -2 , 2 > < 5 , -2 , 2 > ?