Chapter 11.2, Problem 50E

To determine

TO-DETERMINE whether vectors $u=\langle 0,4,-1\rangle$ and $v=\langle 1,0,0\rangle$ are orthogonal, parallel or neither.

Answer to Problem 50E

Vectors $u=\langle 0,4,-1\rangle$ and $v=\langle 1,0,0\rangle$ are orthogonal but not parallel.

Explanation of Solution

Given information:

Consider the vectors $u=\langle 0,4,-1\rangle$ and $v=\langle 1,0,0\rangle$ .

Concept used:

The dot product of two nonzero vectors is zero, then the angle between the vectors is 90° and the vectors are orthogonal.

Two nonzero vectors u and v are parallel when there is some scalar c such that $u=cv$ .

Calculations:

In order to find out the vectors are orthogonal or not, compute the dot product:

  $\begin{array}{c}u\cdot v=\langle 0,4,-1\rangle \cdot \langle 1,0,0\rangle \\ =\left(0\times 1\right)+\left(4\times 0\right)+\left(\left(-1\right)\times 0\right)\\ =0\end{array}$

Since, $u\cdot v=0$ , therefore, the vectors are orthogonal.

In order to find out the vectors are parallel or not, let there be some scalar c such that $u=cv$ , then,

  $\begin{array}{c}\langle 0,4,-1\rangle =c\langle 1,0,0\rangle \\ =\langle c,0,0\rangle \end{array}$

By comparing the coefficients, since, $4=0$ and $-1=0$ , are not possible, therefore the vectors are not parallel.