Chapter 9.6, Problem 52AYU

To determine

To find: The dot product $v \cdot w$ and the angle between $v$ and $w$.

Answer to Problem 52AYU

Solution:

$v \cdot w = 0; \theta = \frac{\pi }{2}$

Explanation of Solution

Given:

$v = i + j; w = -i + j - k$

Calculation:

Dot product:

$\begin{array}{l}v \cdot w = \left(i + j\right) \cdot \left(-i + j - k\right)\\ v \cdot w = 1 \cdot \left(-1\right) + 1 \cdot 1 + 0 \cdot \left(-1\right)\\ v \cdot w = -1 + 1 + 0\\ v \cdot w = 0\end{array}$

Angle $\theta$ between $v$ and $w$:

$\begin{array}{l}\text{cos}\theta = \frac{v\cdot w}{\Vert v\cdot \Vert w} = \frac{0}{\sqrt{1^2 + 1^2}\cdot \sqrt{{\left(-1\right)}^2 + 1^2 + {\left(-1\right)}^2}} = \frac{0}{\sqrt{2}\cdot \sqrt{3}} = 0\\ \theta = {\text{cos}}^{ }0 = \frac{\pi }{2}\end{array}$