Chapter 9, Problem 20RE

To determine

To find: Whether the given two vectors are orthogonal or not if not, then find the angle between them.

Answer to Problem 20RE

The two given vectors are orthogonal.

Explanation of Solution

Given information:

The given vectors are:

  $\begin{array}{l}u=\langle \text{1,}-\text{1}\rangle =i-j\\ v=\langle 1,1\rangle =i+j\end{array}$

Concept used:

The two vectors $u=\langle a,b\rangle$ & $v=\langle c,d\rangle$ are orthogonal if,

  $u\cdot v=0$

Angle between two vectors $u=\langle a,b\rangle$ & $v=\langle c,d\rangle$ is given,

  $\cos \theta =\frac{u\cdot v}{\left|u\right|\times \left|v\right|}$

The dot product of given vectors $u$ & $v$ is calculated as,

  $\begin{array}{c}u\cdot v=1\times 1+\left(-1\right)\times 1\\ =1-1\\ =0\end{array}$

As, $u\cdot v=\text{0}$ , so $u$ & $v$ are orthogonal,

Therefore, the two given vectors are orthogonal.