Chapter 10, Problem 36RE

To determine

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

Answer to Problem 36RE

Thegiven vectors $u$ and $v$ are neither parallel nororthogonal.

Explanation of Solution

Given information:

The given vectors are $u=3j+2k$ and $v=12i-18k$ .

Calculation:

It is known that two vectors are parallel for some scalar $c$ if $u=cv$ .

The component form of vectors $u$ and $v$ are:

  $\begin{array}{l}u=\langle 0,3,2\rangle \\ v=\langle 12,0,-18\rangle \end{array}$

The above two vectors $u$ and $v$ are not parallel to each other because $u\ne cv$ for some scalar $c$ .

To check whether the vector are orthogonal or not, first compute the dot product of $u$ and $v$ . If dot product of vectors is $0$ then $u$ and $v$ will be orthogonal to each.

The dot product can be calculated as:

  $\begin{array}{c}\left(u\cdot v\right)=\langle 0,3,2\rangle \cdot \langle 12,0,-18\rangle \\ =\left(0\right)\left(12\right)+3\left(0\right)+\left(2\right)\left(-18\right)\\ =0+0-36\\ =-36\end{array}$

Therefore, the given vectors $u$ and $v$ are neither parallel nor orthogonal.