Coplanar Vectors

An Intuitive Understanding of Coplanarity

The Necessary and Sufficient Condition for Coplanarity

Methods for Testing Coplanarity

1. The scalar triple product method: compute $\vec{a} \cdot (\vec{b} \times \vec{c})$; if it is zero, they are coplanar
2. The linear-combination method: check whether there exist $k, l, m$ not all zero with $k\vec{a} + l\vec{b} + m\vec{c} = \vec{0}$
3. The determinant method: compute the determinant formed by the coordinates of the three vectors; if it is zero, they are coplanar
4. The volume method: if the parallelepiped spanned by the three vectors has zero volume, they are coplanar

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Worked Examples and Applications

A positive example: coplanar vectors

Example 1: decide whether the vectors $\vec{a} = (1, 0, 0)$, $\vec{b} = (0, 1, 0)$ and $\vec{c} = (1, 1, 0)$ are coplanar.

Solution:

1. The scalar triple product: $\vec{a} \cdot (\vec{b} \times \vec{c})$

   • $\vec{b} \times \vec{c} = (0, 1, 0) \times (1, 1, 0) = (0, 0, -1) = (0, 0, -1)$
   • $\vec{a} \cdot (0, 0, -1) = 1 \cdot 0 + 0 \cdot 0 + 0 \cdot (-1) = 0$

2. Or the linear combination: look for $k, l, m$ with $k(1,0,0) + l(0,1,0) + m(1,1,0) = (0,0,0)$

   • the system: $k + m = 0$, $l + m = 0$, $0 = 0$
   • solving gives $k = 1$, $l = 1$, $m = -1$ (the coefficients are not all zero)

A negative example: non-coplanar vectors

Example 2: decide whether the vectors $\vec{a} = (1, 0, 0)$, $\vec{b} = (0, 1, 0)$ and $\vec{c} = (0, 0, 1)$ are coplanar.

Solution:

1. The scalar triple product: $\vec{a} \cdot (\vec{b} \times \vec{c})$

   • $\vec{b} \times \vec{c} = (0, 1, 0) \times (0, 0, 1) = (1, 0, 0)$
   • $\vec{a} \cdot (1, 0, 0) = 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 = 1 ≠ 0$

2. Or the determinant:

   $$\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} = 1 ≠ 0$$

An example of linear dependence

Example 3: decide whether the vectors $\vec{a} = (1, 2, 3)$, $\vec{b} = (2, 4, 6)$ and $\vec{c} = (3, 6, 9)$ are coplanar.

Solution:

1. Observe the relation: $\vec{b} = 2\vec{a}$, $\vec{c} = 3\vec{a}$
2. The linear combination: $1\vec{a} + (-2)\vec{b} + 1\vec{c} = (0,0,0)$
3. The coefficients $(1, -2, 1)$ are not all zero

Application Scenarios

Applications in spatial geometry

1. Equations of planes: the plane determined by three points in space can be described by using coplanar vectors to represent the position vector of any point on the plane
2. Linear dependence: coplanarity of vectors is an important concept in studying linear dependence
3. Solid geometry: coplanar vectors play an important role in solving problems in spatial geometry

A reminder of common misconceptions

Practice

Exercise 1

Decide whether the vectors $\vec{a} = (1, 1, 0)$, $\vec{b} = (1, 0, 1)$ and $\vec{c} = (0, 1, 1)$ are coplanar.

Exercise 2

Decide whether the vectors $\vec{a} = (1, 3, 2)$, $\vec{b} = (2, 6, 4)$ and $\vec{c} = (3, 9, 6)$ are coplanar.

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Summary

Symbols Used in This Article