## 17Calculus - Vector Cross Product Application - Triple Vector Product

##### 17Calculus

The triple vector product (or vector triple product, as it is sometimes called) is so named because the result is a vector. [For comparison, see the triple scalar product.]

When you have three vectors, $$\vec{u}$$, $$\vec{v}$$ and $$\vec{w}$$, the triple vector product is defined as $$\vec{u} \times \vec{v} \times \vec{w}$$.

Practice

Given $$\vec{A}=\hat{i}-2\hat{j}+2\hat{k}$$, $$\vec{B} = 3\hat{i}-\hat{j}-\hat{k}$$ and $$\vec{C}=-\hat{i}-\hat{k}$$, calculate the triple vector product $$\vec{A}\times\vec{B}\times\vec{C}$$.

Problem Statement

Given $$\vec{A}=\hat{i}-2\hat{j}+2\hat{k}$$, $$\vec{B} = 3\hat{i}-\hat{j}-\hat{k}$$ and $$\vec{C}=-\hat{i}-\hat{k}$$, calculate the triple vector product $$\vec{A}\times\vec{B}\times\vec{C}$$.

$$\vec{A}\times\vec{B}\times\vec{C}=-7\hat{i}-\hat{j}+7\hat{k}$$

Problem Statement

Given $$\vec{A}=\hat{i}-2\hat{j}+2\hat{k}$$, $$\vec{B} = 3\hat{i}-\hat{j}-\hat{k}$$ and $$\vec{C}=-\hat{i}-\hat{k}$$, calculate the triple vector product $$\vec{A}\times\vec{B}\times\vec{C}$$.

Solution

Let's work left to right.
$$\vec{A} \times \vec{B} = (\hat{i}-2\hat{j}+2\hat{k}) \times (3\hat{i}-\hat{j}-\hat{k}) =$$ $$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 2 \\ 3 & -1 & -1 \end{vmatrix} =$$ $$(2-(-2))\hat{i} - (-1-6)\hat{j} + (-1-(-6))\hat{k} =$$ $$4\hat{i} + 7\hat{j} +5\hat{k} = \vec{v}$$
Now we need to calculate $$\vec{v} \times \vec{C}$$.
$$(4\hat{i} + 7\hat{j} +5\hat{k}) \times (-\hat{i}-\hat{k}) =$$ $$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 7 & 5 \\ -1 & 0 & -1 \end{vmatrix} =$$ $$(-7-0)\hat{i} - (-4-(-5))\hat{j} +(0-(-7))\hat{k} =$$ $$-7\hat{i} -\hat{j} +7\hat{k}$$

$$\vec{A}\times\vec{B}\times\vec{C}=-7\hat{i}-\hat{j}+7\hat{k}$$

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