Answer to Question #178939 in Calculus for John withers

Question

Determine the angle between the following two vectors.

|𝑔| = (1,2,−3)

|h| = (0,−3,4)

Answer 1

Let $\theta$ be the angle between the vectors $\vec{g}$ and $\vec{h}$ .

Using the transformed dot product equation,

$\theta = \cos^{-1} \left( \frac{ \vec{g}\cdot \vec{h}}{|\vec{g}| |\vec{h}| } \right)$

where

$\vec{g}\cdot \vec{h} = (1)(0) + (2)(-3)+(-3)(4)=-18$ (the dot product of $\vec{g}$ and $\vec{h}$ )

and

$|\vec{g}| = \sqrt{(1)^2+(2)^2+(-3)^2} = \sqrt{14}$ (magnitude of vector $\vec{g}$ )

$|\vec{h}| = \sqrt{(0)^2+(-3)^2+(4)^2} = \sqrt{25}=5$ (magnitude of vector $\vec{h}$ )

Therefore,

$$\theta = \cos^{-1} \left( \frac{-18}{\sqrt{14} (5)} \right)$$

$$\theta = \cos^{-1} \left( -0.9621 \right)$$

$$\theta = 164.18^{0}$$