Exercise 3.3 — Question 9

Topic: Scalar Triple Product — Volume and Coplanarity

This question applies the scalar triple product to find the volume of a parallelepiped or tetrahedron, or to test whether three vectors are coplanar.

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Key Concepts

Scalar Triple Product

For three vectors $\mathbf{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, $\mathbf{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, and $\mathbf{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$, the scalar triple product is:

$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

Volume of a Parallelepiped

The volume of a parallelepiped determined by three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ is:

$V=|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|$

Volume of a Tetrahedron

The volume of a tetrahedron determined by three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ from a common vertex is:

$V=\frac{1}{6}|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|$

Condition for Coplanarity

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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Method

Step 1: Write the three vectors in component form.

Step 2: Set up the $3\times 3$ determinant with the components of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ as rows.

Step 3: Evaluate the determinant by expanding along the first row:

$\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=a_1(b_2{c}_3-b_3{c}_2)-a_2(b_1{c}_3-b_3{c}_1)+a_3(b_1{c}_2-b_2{c}_1)$

Step 4:

• For volume of parallelepiped: $V=|\text{result}|$
• For volume of tetrahedron: $V=\frac{1}{6}|\text{result}|$
• For coplanarity: check if result $=0$