What is the difference between and the Cartesian plane equation?

Substituting into gives the Cartesian form , which can be rearranged to with .

Vectors (Edexcel IAL Further Maths FP3)

Edexcel International A-Level Further Mathematics· FP3 5.1, 5.2, 5.3 (2018 Spec, Issue 3)· 25 min read

1. Vector Cross Product & Triple Scalar Product★★★☆☆⏱ 8 min

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📘 Definition

Vector (Cross) Product

$a \times b$

Vector perpendicular to both $a$ and $b$ , with magnitude $∣ a ∣∣ b ∣ sin θ$ (where $θ$ is the angle between the vectors) and direction given by the right-hand rule. Has the anti-commutative property $a \times b = - (b \times a)$ .

Example:

For $a = 2 i + j$ , $b = i - 3 j + k$ , $a \times b = i - 2 j - 7 k$ .

The magnitude $∣ a \times b ∣$ equals the area of the parallelogram formed by $a$ and $b$ as adjacent sides. Divide this by 2 to get the area of the triangle formed by the two vectors.

📘 Definition

Triple Scalar Product

$a \cdot (b \times c)$

Scalar result of the dot product of $a$ with the cross product of $b$ and $c$ , equal to the determinant of the 3×3 matrix formed by the components of $a, b, c$ . Its magnitude gives the volume of the parallelepiped formed by the three vectors as adjacent edges. Obeys the cyclic identity: $a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)$ .

📐 Worked Example

Find the volume of the parallelepiped with edges $a = i + 2 j$ , $b = 2 j + k$ , $c = i - j + 3 k$ .

1
1. Compute the cross product $b \times c$ using the determinant formula:
2
$\boldsymbol{b} \times \boldsymbol{c} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ 0 & 2 & 1 \\ 1 & -1 & 3 \end{vmatrix} = 7\boldsymbol{i} + \boldsymbol{j} - 2\boldsymbol{k}$
3
1. Take the dot product of $a$ with the result:
4
$\boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c}) = (1)(7) + (2)(1) + (0)(-2) = 9$
5
1. The magnitude of the result is the volume: 9 cubic units.

Exam tip:

Sign from cross product only matters for orientation questions: take the magnitude when calculating area or volume to avoid errors.

2. Equations of Planes★★★☆☆⏱ 7 min

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There are two standard forms of plane equations you will need to convert between and apply for FP3 problems, both with equivalent Cartesian forms.

📘 Definition

Normal (Scalar) Form of a Plane

$r \cdot n = p$

Where $r$ is the position vector of any point on the plane, $n$ is the normal vector to the plane, and $p = a \cdot n$ for a known point $a$ on the plane. Equivalent Cartesian form: $n_{1} x + n_{2} y + n_{3} z = p$ , or $n_{1} x + n_{2} y + n_{3} z - p = 0$ with $d = - p$ .

📘 Definition

Parametric Form of a Plane

$r = a + s b + t c$

Where $a$ is the position vector of a fixed point on the plane, $b$ and $c$ are non-parallel direction vectors lying on the plane, and $s, t$ are scalar parameters.

📐 Worked Example

Find the normal form and Cartesian form of the plane passing through $(1, 2, 0)$ with direction vectors $b = i - j + k$ and $c = 2 i + 3 j$ .

1
1. Compute the normal vector as the cross product of the two direction vectors:
2
$\boldsymbol{n} = \boldsymbol{b} \times \boldsymbol{c} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ 1 & -1 & 1 \\ 2 & 3 & 0 \end{vmatrix} = -3\boldsymbol{i} + 2\boldsymbol{j} +5\boldsymbol{k}$
3
1. Calculate $p = a \cdot n$ where $a = i + 2 j$ :
4
$p = (1)(-3) + (2)(2) + (0)(5) = 1$
5
1. Normal form: $r \cdot (- 3 i + 2 j + 5 k) = 1$ (multiply by -1 for positive leading coefficient if preferred).
6
1. Cartesian form: Replace $r$ with $x i + y j + z k$ to get $3 x - 2 y - 5 z + 1 = 0$ .

Exam tip:

If you are given three points on a plane, find two direction vectors between pairs of points, then compute their cross product to get the normal vector for the plane.

3. 3D Applications: Point-Plane Distance & Plane Intersection★★★★☆⏱ 6 min

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The first two core 3D applications for FP3 use plane equations and vector products to solve distance and intersection problems. The perpendicular distance formula for a point to a plane is provided in your exam formula booklet.

📐 Worked Example

Find the perpendicular distance of the point $(2, - 1, 3)$ from the plane $3 x + 2 y - z + 4 = 0$ .

1
1. Identify values for the distance formula: $n_{1} = 3, n_{2} = 2, n_{3} = - 1, d = 4, α = 2, β = - 1, γ = 3$ .
2
1. Compute the absolute value of the numerator:
3
$|3(2) + 2(-1) + (-1)(3) + 4| = |6 - 2 - 3 + 4| = 5$
4
1. Compute the magnitude of the normal vector:
5
$\sqrt{3^2 + 2^2 + (-1)^2} = \sqrt{14}$
6
1. Distance = $\frac{5}{14}$ , or rationalized to $\frac{5 14}{14}$ .

The line of intersection of two planes is parallel to the cross product of the normal vectors of the two planes, as it is perpendicular to both normals. Find one point lying on both planes to define the full line.

📐 Worked Example

Find the line of intersection of planes $2 x - y + z = 5$ and $x + y - z = 1$ , in the form $(r - a) \times b = 0$ .

1
1. Extract normal vectors: $n_{1} = 2 i - j + k$ , $n_{2} = i + j - k$ .
2
1. Compute the direction vector of the line as the cross product of the normals:
3
$\boldsymbol{b} = \boldsymbol{n_1} \times \boldsymbol{n_2} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ 2 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix} = 3\boldsymbol{j} + 3\boldsymbol{k} = 3(\boldsymbol{j} + \boldsymbol{k})$
4
1. Find a point on both planes: set $x = 2$ , then solve to get $y = 0, z = 1$ , so $a = 2 i + 0 j + k$ .
5
1. Final line equation: $(r - (2 i + k)) \times (j + k) = 0$ .

Exam tip:

If setting $x = 0$ to find a point on the intersection of two planes leads to inconsistent equations, try setting $y = 0$ or $z = 0$ instead.

4. 3D Applications: Shortest Distance Between Skew Lines★★★★★⏱ 7 min

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📘 Definition

Skew Lines

Two lines in 3D space that are neither parallel nor intersecting. The shortest distance between them is the length of the line segment that is perpendicular to both lines.

The formula for the shortest distance between skew lines $L_{1} : r = a + λ u$ and $L_{2} : r = b + μ v$ is $\frac{∣ ( b - a ) \cdot ( u \times v ) ∣}{∣ u \times v ∣}$ . The numerator is the volume of the parallelepiped formed by $(b - a), u, v$ , divided by the area of the base parallelogram to give the perpendicular height (distance).

📐 Worked Example

Find the shortest distance between skew lines $L_{1} : r = i + 2 j + k + λ (2 i - j + 3 k)$ and $L_{2} : r = 2 i - j + 2 k + μ (3 i + j - k)$ .

1
1. Identify values: $a = i + 2 j + k$ , $u = 2 i - j + 3 k$ , $b = 2 i - j + 2 k$ , $v = 3 i + j - k$ .
2
1. Compute the vector between the fixed points of the two lines:
3
$\boldsymbol{b} - \boldsymbol{a} = \boldsymbol{i} - 3\boldsymbol{j} + \boldsymbol{k}$
4
1. Compute the cross product of the direction vectors:
5
$\boldsymbol{u} \times \boldsymbol{v} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ 2 & -1 & 3 \\ 3 & 1 & -1 \end{vmatrix} = -2\boldsymbol{i} + 11\boldsymbol{j} +5\boldsymbol{k}$
6
1. Compute the absolute value of the triple scalar product:
7
$|(\boldsymbol{b} - \boldsymbol{a}) \cdot (\boldsymbol{u} \times \boldsymbol{v})| = |-2 -33 +5| = 30$
8
1. Compute the magnitude of the cross product:
9
$|\boldsymbol{u} \times \boldsymbol{v}| = \sqrt{(-2)^2 + 11^2 + 5^2} = \sqrt{150} = 5\sqrt{6}$
10
1. Shortest distance = $\frac{30}{5 6} = 6$ units.

Exam tip:

Double-check cross product calculations for skew line problems: arithmetic errors here are the most common cause of lost marks on this topic.

5. Common Pitfalls

Wrong move:

Swapping the order of vectors in a cross product and forgetting the sign flips.

Why:

The cross product is anti-commutative: $a \times b = - (b \times a)$ .

Correct move:

Keep track of vector order, or take the magnitude when calculating area/volume so sign does not affect results.

Wrong move:

Assuming non-intersecting 3D lines are parallel.

Why:

3D space has an extra dimension, so lines can be non-parallel and non-intersecting (skew).

Correct move:

First check if direction vectors are scalar multiples to confirm parallelism, before checking for intersection.

Wrong move:

Using parallel direction vectors for the parametric plane equation $r = a + s b + t c$ .

Why:

Parallel vectors do not define a unique plane, as they only span a line.

Correct move:

Use two non-parallel direction vectors, verified by a non-zero cross product.

Wrong move:

Forgetting the absolute value in distance formulas.

Why:

Distance is a non-negative scalar; sign only indicates orientation relative to the plane normal.

Correct move:

Always include absolute value around the numerator in point-plane and skew line distance calculations.

Wrong move:

Assuming $p$ in $r \cdot n = p$ is the perpendicular distance from the origin.

Why:

This is only true if $n$ is a unit normal vector.

Correct move:

If you need the origin distance, divide $p$ by the magnitude of $n$ , or use the Cartesian distance formula for point $(0, 0, 0)$ .

6. Quick Reference Cheatsheet

Concept	Formula/Rule	Use Case
Cross Product	$∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$	Find area of parallelogram/triangle, plane normal vectors
Triple Scalar Product	$∣ a \cdot (b \times c) ∣$	Volume of parallelepiped, skew line distance calculations
Plane: Normal Form	$r \cdot n = p$	Point-plane distance checks, verifying points lie on a plane
Plane: Parametric Form	$r = a + s b + t c$	Defining planes from a point and two direction vectors
Point-Plane Distance	$\frac{∣ n _{1} α + n _{2} β + n _{3} γ + d ∣}{n _{1}^{2} + n _{2}^{2} + n _{3}^{2}}$	Perpendicular distance from a point to a plane
Skew Line Distance	$\frac{∣ ( b - a ) \cdot ( u \times v ) ∣}{∣ u \times v ∣}$	Shortest distance between two non-parallel, non-intersecting lines

7. Frequently Asked

Do I need to memorize the cross product component formula?

No, the exam formula booklet provides the determinant form for the cross product, but you should practice computing it quickly to avoid arithmetic errors in exams.

What is the difference between $r \cdot n = p$ and the Cartesian plane equation?

Substituting $r = x i + y j + z k$ into $r \cdot n = p$ gives the Cartesian form $n_{1} x + n_{2} y + n_{3} z = p$ , which can be rearranged to $n_{1} x + n_{2} y + n_{3} z - p = 0$ with $d = - p$ .

Can I use a calculator to compute cross products and determinants?

Standard scientific calculators are permitted, but CAS calculators are forbidden. You must show full working for cross product and dot product calculations to earn method marks.

Going deeper

official_documentEdexcel IAL Further Maths FP3 SpecificationRefer to page 72 for vectors content
formula_sheetEdexcel FP3 Formula BookletAll core vector formulas are provided in the exam

What's Next

Now that you have mastered FP3 vectors, you can apply these skills to past paper questions, as this topic typically makes up 15-20% of the FP3 exam marks. Vectors are also used in other further math units including FP2 complex number geometry and M3 kinematics, but focus strictly on the pure geometry applications specified for FP3 to avoid off-syllabus content. Practice writing all steps clearly, as method marks are awarded for correct cross and dot product calculations even if you make minor arithmetic errors.