Exercise 3.4 — Question 3: Coplanarity of Vectors

Concept Overview

Three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are said to be coplanar if they all lie in the same plane. The algebraic condition for coplanarity uses the scalar triple product.

Condition for Coplanarity

Three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are coplanar if and only if their scalar triple product equals zero:

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

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 the condition becomes:

$\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=0$

Geometric reason: The scalar triple product gives the volume of the parallelepiped formed by the three vectors. If the vectors are coplanar, the parallelepiped is flat and has zero volume.

Scalar Triple Product — Determinant Form

For vectors $\mathbf{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, $\mathbf{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, $\mathbf{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$:

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

Worked Example

Problem: Find the value of $d$ for which the vectors $\mathbf{a}=\hat{i}+2\hat{j}+3\hat{k},\mathbf{b}=2\hat{i}-\hat{j}+\hat{k},\mathbf{c}=3\hat{i}+d\hat{j}+2\hat{k}$ are coplanar.

Solution: Set the scalar triple product equal to zero:

$\left|\begin{array}{ccc}1& 2& 3\\ 2& -1& 1\\ 3& d& 2\end{array}\right|=0$

Expanding along the first row: $1[(-1)(2)-(1)(d)]-2[(2)(2)-(1)(3)]+3[(2)(d)-(-1)(3)]=0$ $1[-2-d]-2[4-3]+3[2d+3]=0$ $-2-d-2+6d+9=0$ $5d+5=0$ $d=-1$

Verification: Substituting $d=-1$ back into the determinant gives zero, confirming the vectors are coplanar.

Volume of a Parallelepiped

The volume of the parallelepiped with coterminal edges $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ is:

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

The volume of the tetrahedron formed by the same three vectors is:

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