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: $u=\frac{3}{4}i-\frac{1}{2}j+2k$ and $v=4i+10j+k$

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,

  $\begin{array}{l}u.v=\left(\frac{3}{4}i-\frac{1}{2}j+2k\right).\left(4i+10j+k\right)\\ u.v=\frac{3}{4}\left(4\right)+\left(-\frac{1}{2}\right)\left(10\right)+2\left(1\right)=3-5+2=0\end{array}$

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

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