Question

if u=(2,-3,4) & v=(-1,2,0)& w=(5,-1,2) find the angle b/n (u-2v) & (u+2w)

Answer 1

$(u-2v) = (2,-3,4) - 2\cdot(-1,2,0) = (4,-7,4)$

$(u+2w) = (2,-3,4) + 2\cdot(5,-1,2) = (12, -5, 8)$

The angle $\alpha$ between two vectors $\vec{a}$ and $\vec{b}$ can be obtained as $\cos\alpha = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}.$

$\cos\alpha = \dfrac{4\cdot12+(-7)\cdot(-5)+4\cdot8}{\sqrt{4^2+(-7)^2+4^2}\sqrt{12^2+(-5)^2+8^2}},\\ \cos\alpha = \dfrac{115}{9\cdot\sqrt{233}}, \\ \alpha = 33.2^\circ.$