Question

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

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

Short answer

The expression$\left|\left|v-w\right|\right|=\sqrt{38}.$

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,

$$\begin{gathered} \left|\left|v-w\right|\right| \\ =\left|\left|\left(3i+j-2k\right)-\left(-3i+2j-k\right)\right|\right| \\ =\left|\left|6i-j-k\right|\right| \end{gathered}$$

As we know the length of a vector can be found as $$\begin{gathered} \left|\left|v\right|\right|=\left|\left|ai+bj+ck\right|\right| \\ \left|\left|v\right|\right|=\sqrt{a^2+b^2+c^2} \end{gathered}$$

So,

$$\begin{gathered} \left|\left|v-w\right|\right|=\left|\left|6i-j-k\right|\right| \\ \left|\left|v-w\right|\right|=\sqrt{6^2+{\left(-1\right)}^2+{\left(-1\right)}^2} \\ \left|\left|v-w\right|\right|=\sqrt{38} \end{gathered}$$