1.6 Vectors

2026 Syllabus Objectives

Use the equation of a plane in any of the forms $ax + by + cz = d$ or $\mathbf{r} \cdot \mathbf{n} = p$ or $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$ and convert equations of planes from one form to another as necessary in solving problems
Recall that the vector product $\mathbf{a} \times \mathbf{b}$ of two vectors can be expressed either as $|\mathbf{a}||\mathbf{b}|\sin \theta \, \hat{\mathbf{n}}$ , where $\hat{\mathbf{n}}$ is a unit vector, or in component form as $(a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$
Use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections, including:
- Determining whether a line lies in a plane, is parallel to a plane or intersects a plane, and finding the point of intersection of a line and a plane when it exists
- Finding the foot of the perpendicular from a point to a plane
- Finding the angle between a line and a plane, and the angle between two planes
- Finding an equation for the line of intersection of two planes
- Calculating the shortest distance between two skew lines
- Finding an equation for the common perpendicular to two skew lines

🔑 Key Concepts and Definitions

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Scalar Product

The scalar product (or dot product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as:

$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos \theta$

where $\theta$ is the angle between the two vectors.

Vector Product

The vector product (or cross product) of two vectors results in a vector that is perpendicular to the plane containing the two original vectors. It is a fundamental tool for finding normal vectors to planes and solving geometric problems.

Unit Vector

A unit vector is a vector with a magnitude of 1. It is often used to indicate direction without considering magnitude.

Common Perpendicular

A line that is perpendicular to two skew lines (non-parallel, non-intersecting lines) is called a common perpendicular.

📐 Equation of a Plane

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A plane in three-dimensional space can be represented in several different forms. Understanding these forms and being able to convert between them is essential for solving problems involving planes.

Form 1: Cartesian Form

$ax + by + cz = d$

where $a$ , $b$ , $c$ , and $d$ are constants, and $(a, b, c)$ represents the normal vector to the plane.

Form 2: Vector Form (Scalar Product)

$\mathbf{r} \cdot \mathbf{n} = p$

where:

$\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ is the position vector of a general point on the plane
$\mathbf{n}$ is the normal vector to the plane
$p$ is a scalar constant

Form 3: Parametric Form

$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$

where:

$\mathbf{a}$ is the position vector of a known point on the plane
$\mathbf{b}$ and $\mathbf{c}$ are two non-parallel direction vectors lying in the plane
$\lambda$ and $\mu$ are scalar parameters

Converting Between Forms

Converting from Cartesian to Vector Form:

If the plane is given as $ax + by + cz = d$ , then the normal vector is $\mathbf{n} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ , and the vector form is $\mathbf{r} \cdot (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = d$ .

Converting from Vector Form to Cartesian:

If $\mathbf{r} \cdot \mathbf{n} = p$ where $\mathbf{n} = n_1\mathbf{i} + n_2\mathbf{j} + n_3\mathbf{k}$ , then substituting $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ gives $n_1x + n_2y + n_3z = p$ .

Converting from Parametric to Cartesian:

Given $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$ , find the normal vector $\mathbf{n}$ by computing the vector product $\mathbf{b} \times \mathbf{c}$ . Then use a known point on the plane to find the constant.

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