Chapter 11.2, Problem 51E

To determine

To find: Whether the vectors $u$ and $v$ are orthogonal, parallel , or neither.

Answer to Problem 51E

The vectors $u$ and $v$ are orthogonal.

Explanation of Solution

Given vectors: $u=-2i+3j-k$ and $v=2i+j-k$

Calculation:

We know that two vectors are orthogonal if and only if their scalar product is zero and are parallel if and only if they are scalar multiple of one another.

Now,

  $\begin{array}{l}u.v=\left(-2i+3j-k\right).\left(2i+j-k\right)\\ u.v=-2\left(2\right)+3\left(1\right)-1\left(-1\right)=-4+3+1=0\end{array}$

That is the scalar product of vectors $u$ and $v$ is 0.

Therefore, the vectors $u$ and $v$ are orthogonal.