Question

In Problems 7–16,

(a) find the dot product $v·w$;

(b) find the angle between v and w;

(c) state whether the vectors are parallel, orthogonal, or neither.

$$\begin{gathered} v=2i+j \\ w=i-2j \end{gathered}$$

Short answer

Part a.$v·w=0$

Part b. The angle between the vectors is $90°$.

Part c. The vectors are orthogonal.

Step-by-step solution

Part (a) Step 1. Given Information

Given two vectors $v=2i+j$and $w=i-2j$.

We need to find their dot product, the angle between the two vectors, and the nature of the two vectors.

Part (a) Step 2. Find the dot product   

The dot product for the given vectors is given as

$$\begin{gathered} v·w=(2i+j)·(i-2j) \\ v·w=2·1+1·(-2) \\ v·w=2-2 \\ v·w=0 \end{gathered}$$

Part (b) Step 1. Find the angle   

The angle between the vectors v and w is

$\begin{array}{rcl}\cos \theta & =& \frac{v·w}{\left|v\right|\left|w\right|}\end{array}$

Now as $v·w=0$so we get

$\begin{array}{rcl}\cos \theta & =& \frac{0}{\left|v\right|\left|w\right|}\\ \cos \theta & =& 0\\ \theta & =& 90°\end{array}$

Part (c) Step 1. Identify the nature of the vectors   

As the angle between the two vectors is $90°$, so the given two vectors v andw are othrogonal.