Exercise 3.3 — Question 15

Problem

Find the volume of the parallelepiped determined by the vectors: $\mathbf{a}=\hat{i}+2\hat{j}-\hat{k},\mathbf{b}=2\hat{i}-\hat{j}+3\hat{k},\mathbf{c}=3\hat{i}+\hat{j}+2\hat{k}$

Also determine whether the vectors are coplanar.

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Key Concepts

Scalar Triple Product

The scalar triple product of three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ is defined as: $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$

In determinant form, if $\mathbf{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k},\mathbf{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k},\mathbf{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ then: $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

Volume of a Parallelepiped

The volume of a parallelepiped with edges along $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ is: $V=|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|$

Volume of a Tetrahedron

The volume of a tetrahedron formed by three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ from a common vertex is: $V_{\text{tetrahedron}}=\frac{1}{6}|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|$

Coplanar Vectors

Three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are coplanar if and only if their scalar triple product is zero: $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=0$

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Solution

Step 1: Set up the determinant.

With $\mathbf{a}=\hat{i}+2\hat{j}-\hat{k}$, $\mathbf{b}=2\hat{i}-\hat{j}+3\hat{k}$, $\mathbf{c}=3\hat{i}+\hat{j}+2\hat{k}$:

$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\left|\begin{array}{ccc}1& 2& -1\\ 2& -1& 3\\ 3& 1& 2\end{array}\right|$

Step 2: Expand along the first row.

$=1\cdot \left|\begin{array}{cc}-1& 3\\ 1& 2\end{array}\right|-2\cdot \left|\begin{array}{cc}2& 3\\ 3& 2\end{array}\right|+(-1)\cdot \left|\begin{array}{cc}2& -1\\ 3& 1\end{array}\right|$

$=1(-2-3)-2(4-9)+(-1)(2+3)$

$=1(-5)-2(-5)+(-1)(5)$

$=-5+10-5=0$

Step 3: Interpret the result.

Since $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=0$:

• Volume of parallelepiped $=|0|=0$ cubic units
• The vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are coplanar (they lie in the same plane).

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Summary