6.1.1 Equations of Lines in 3D

Introduction

In 3D space, a straight line can be described in multiple forms:

Vector Form:

{r} = \mathbf{a} + \lambda \mathbf{b}

Cartesian Form:

\frac{x-a_1}{b_1} = \frac{y-a_2}{b_2} = \frac{z-a_3}{b_3}

Both forms are equivalent and represent the same line.

Vector Form of a Line

The vector equation of a line is written as:

{r} = \mathbf{a} + \lambda \mathbf{b}

where:

$r$ is the position vector of any point on the line.
$a$ is the position vector of a fixed point on the line.
$b$ is the direction vector of the line.
$\lambda$ is a scalar parameter.

Cartesian Form of a Line

The Cartesian form is derived from the vector form:

\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \implies x = a_1 + \lambda b_1, \; y = a_2 + \lambda b_2, \; z = a_3 + \lambda b_3

Eliminating $\lambda$ gives:

\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}

Worked Examples

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Example 1: Find the Vector Equation of a Line

Find the vector equation of the line passing through $(1, 2, 3)$ with direction vector ${b} = (4, -1, 2)$

Step 1: Identify components:

Position vector of a point:

{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}

Direction vector:

{b} = \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}

Step 2: Write the vector equation:

\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}

Result:

\mathbf{r} = \begin{pmatrix} 1 + 4\lambda \\ 2 - \lambda \\ 3 + 2\lambda \end{pmatrix}

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Example 2: Convert Vector Form to Cartesian Form

Convert

\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}

into Cartesian form.

Step 1: Extract components:

$x = 2 + \lambda$
$y = 1 - 2\lambda$
$z = -3 + 4\lambda$

Step 2: Eliminate $\lambda:$

\lambda = x - 2, \quad \lambda = \frac{1-y}{2}, \quad \lambda = \frac{z+3}{4}

Step 3: Combine into Cartesian form:

\frac{x-2}{1} = \frac{1-y}{-2} = \frac{z+3}{4}

Result:

\frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{4}

Note Summary

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Common Mistakes:

Confusing the position and direction vectors: Ensure $\mathbf{a}$ is a fixed point on the line and $\mathbf{b}$ is the direction vector.
Failing to eliminate $\lambda$ properly in Cartesian form: Carefully solve for $\lambda$ in all three coordinates.
Mixing up vector components: Write direction vectors clearly to avoid errors in simultaneous equations.
Not checking solutions: Substitute solutions back into both line equations to confirm consistency.