Question

In Problems 40–45, use the vectors $v=3i+j-2k$and $w=-3i+2j-k$to find each expression.

$\left|\left|v\right|\right|-\left|\left|w\right|\right|$

Short answer

The expression$\left|\left|v\right|\right|-\left|\left|w\right|\right|=0.$

Step-by-step solution

Step 1. Given Information

The given vectors are$v=3i+j-2kandw=-3i+2j-k.$

Step 2. Finding the given expression 

To find the expression, substitute the values of vectors in the given expression.

Thus, $\left|\left|v\right|\right|-\left|\left|w\right|\right|=\left|\left|3i+j-2k\right|\right|-\left|\left|-3i+2j-k\right|\right|$

As we know the length of a vector can be found as

$$\begin{gathered} \left|\left|v\right|\right|=\left|\left|ai+bj+ck\right|\right| \\ \left|\left|v\right|\right|=\sqrt{a^2+b^2+c^2} \end{gathered}$$

Step 3. Solve

By proceeding with the calculation further

$$\begin{gathered} \left|\left|v\right|\right|-\left|\left|w\right|\right| \\ =\left|\left|3i+j-2k\right|\right|-\left|\left|-3i+2j-k\right|\right| \\ =\sqrt{3^2+1^2+{\left(-2\right)}^2}-\sqrt{{\left(-3\right)}^2+2^2+{\left(-1\right)}^2} \\ =\sqrt{14}-\sqrt{14} \\ =0 \end{gathered}$$