Question

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

$v·\left(v\times w\right)$

Short answer

The expression$v·\left(v\times w\right)=0.$

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, $v·\left(v\times w\right)=\left(3i+j-2k\right)·\left(\left(3i+j-2k\right)\times \left(-3i+2j-k\right)\right).$

Now, let's first find the cross product, by using the definition of it.

If $v=a_1\mathbf{i}+{\mathrm{b}}_1\mathbf{j}+{\mathrm{c}}_1\mathbf{k}$and $w=a_2\mathbf{i}+{\mathrm{b}}_2\mathbf{j}+{\mathrm{c}}_2\mathbf{k}$are two vectors in space, the

cross product $v\times w$is defined as the vector

$v\times w=\left(b_1{c}_2-b_2{c}_1\right)\mathbf{i}\mathbf{}-\left({\mathrm{a}}_1{\mathrm{c}}_2-{\mathrm{a}}_2{\mathrm{c}}_1\right)\mathbf{j}\mathbf{}+\left({\mathrm{a}}_1{\mathrm{b}}_2-{\mathrm{a}}_2{\mathrm{b}}_1\right)\mathbf{k}\mathbf{.}$

Step 3. Solve 

By proceeding with the calculation further using the formula of the cross product.

So,

$$\begin{gathered} v\times w=\left(1·\left(-1\right)-2·\left(-2\right)\right)\mathbf{i}\mathbf{}-\left(3·\left(-1\right)-\left(-3\right)·\left(-2\right)\right)\mathbf{j}\mathbf{}+\left(3·2-\left(-3\right)·1\right)\mathbf{k} \\ \mathrm{v}\times \mathrm{w}=3\mathbf{i}+9\mathbf{j}+9\mathbf{k} \end{gathered}$$

Let's find the dot product, the formula of the dot product is $v·w=a_1{a}_2+b_1{b}_2+c_1{c}_2.$

So,

$$\begin{gathered} v·\left(v\times w\right)=\left(3i+j-2k\right)·\left(3i+9j+9k\right) \\ v·\left(v\times w\right)=\left(3·3\right)+\left(1·9\right)+\left(-2·9\right) \\ v·\left(v\times w\right)=9+9-18 \\ v·\left(v\times w\right)=0 \end{gathered}$$