Chapter 10, Problem 26CR

To determine

To find: the angle between the vectors.

Answer to Problem 26CR

The angle between the vectors is $\frac{\pi }{2}$ .

Explanation of Solution

Given information:

The vectors are $u=\langle 2\sqrt{2},-4,4\rangle ,v=\langle -\sqrt{2},1,2\rangle$ .

Formula used: the angle between vectors,$\cos \theta =\frac{u·v}{\left|u\right|\left|v\right|}$

the dot product of vectors,

  $u·v=u_1·v_1+u_2·v_2+u_3·v_3$

Calculation: to find the angle between vectors, first substitute and then simplify,

  $\begin{array}{l}\cos \theta =\frac{u·v}{\left|u\right|\left|v\right|}\\ \cos \theta =\frac{\left(2\sqrt{2}\right)\left(-\sqrt{2}\right)+\left(-4\right)\left(1\right)+\left(4\right)\left(2\right)}{\sqrt{{\left(2\sqrt{2}\right)}^2+{\left(-4\right)}^2+{\left(4\right)}^2}·\sqrt{{\left(-\sqrt{2}\right)}^2+{\left(1\right)}^2+{\left(2\right)}^2}}\text{ [use the formula and substitute]}\\ \text{cos}\theta =\frac{-4-4+8}{\sqrt{8+16+16}·\sqrt{4+1+4}}\text{ [multiply]}\\ \cos \theta =0\text{ [simplify]}\\ \theta =\frac{\pi }{2}\end{array}$

Thus, The angle between the vectors is $\frac{\pi }{2}$ .