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Edexcel A-level Mathematics (9MA0) Β· Vectors
Mini-Lesson

Vectors

Topic 10 covers vectors in two and three dimensions: column and i, j, k notation, magnitude and direction, adding and scaling vectors, unit vectors, position vectors and the distance between two points in 3D, testing for parallel vectors and collinear points, dividing a line in a given ratio, and using vectors to solve geometric problems.

i, j, k andmagnitudeposition vectors& distancegeometry withvectorsTopic 10: 3D vectors are A-level only β€” and the scalar product is NOT on 9MA0

Work through each screen, answer the questions as you go (some are wordy, most are calculations) and collect ⭐ stars. Watch for the Exam trap notes β€” that is where the marks get thrown away. Press Start when you are ready.

Vectors Β· notation

Vectors in two and three dimensions

A vector has magnitude and direction; a scalar has magnitude only. In 3D we use the mutually perpendicular unit vectors i, j, k along the x, y and z axes.

z (k) y (j) x (i) a = 2i βˆ’ 3j + 6k |a| = √(2Β² + 3Β² + 6Β²) = √49 = 7
A 3D vector and its magnitude β€” Pythagoras, applied twice.
a = xi + yj + zk · |a| = √(x² + y² + z²)the column form is the same thing written vertically

Exam trap: the magnitude is never negative β€” square the components, then take the positive root. |2i βˆ’ 3j + 6k| = √(4 + 9 + 36) = 7, not √(4 βˆ’ 9 + 36).

Calculate

Your turn β€” magnitude in 2D

1Find the magnitude of the vector 3i βˆ’ 4j.
Set-up

|3i βˆ’ 4j| = √(9 + 16) = √25 = 5.

Hint: √(3Β² + (βˆ’4)Β²) = √25.
Calculate

Your turn β€” magnitude in 3D

2Find the magnitude of the vector a = 2i βˆ’ 3j + 6k.
Set-up

|a| = √(4 + 9 + 36) = √49 = 7.

Hint: √(2² + 3² + 6²) = √49.
Quick check

Magnitude in 3D

?What is |xi + yj + zk|?
Vectors Β· algebra

Adding, scaling and unit vectors

  • Adding: add the components. (2i + j) + (3i βˆ’ 4j) = 5i βˆ’ 3j. Geometrically it is the triangle (or parallelogram) law.
  • Scaling: Ξ»a has the same direction as a (opposite if Ξ» < 0) and magnitude |Ξ»||a|.
  • Parallel: b is parallel to a exactly when b = Ξ»a for some scalar Ξ». So 4i βˆ’ 2j + 6k = 2(2i βˆ’ j + 3k) is parallel to 2i βˆ’ j + 3k.
  • Unit vector: divide by the magnitude β€” Γ’ = a / |a|, which has magnitude 1.
Worked example β€” the unit vector in the direction of a = 2i βˆ’ 3j + 6k

|a| = 7 (previous screen).

Γ’ = (1/7)(2i βˆ’ 3j + 6k) = (2/7)i βˆ’ (3/7)j + (6/7)k.

The i-component is 2/7 = 0.2857 (4 d.p.). Check: (2/7)Β² + (3/7)Β² + (6/7)Β² = (4 + 9 + 36)/49 = 1 βœ“

Calculate

Your turn β€” unit vector

3Find the i-component of the unit vector in the direction of a = 2i βˆ’ 3j + 6k. (Give a decimal to 4 d.p., or the fraction 2/7.)
Set-up

Γ’ = (1/7)(2i βˆ’ 3j + 6k), so the i-component is 2/7 = 0.2857 (4 d.p.).

Hint: Divide each component by |a| = 7.
Quick check

Parallel vectors

?Which vector is parallel to 2i βˆ’ j + 3k?
Vectors Β· direction

Magnitude and direction form

In 2D you can convert between component form and magnitude–direction form.

|v| = √(xΒ² + yΒ²) Β· tan ΞΈ = y/xΞΈ is measured anticlockwise from the positive x-axis β€” sketch it to get the correct quadrant
Worked example β€” v = 3i + 4j

|v| = √(9 + 16) = 5.

ΞΈ = arctan(4/3) = 53.1Β° (1 d.p.) above the positive x-axis. Both components are positive, so the vector really is in the first quadrant. βœ“

Exam trap: arctan only ever returns an angle between βˆ’90Β° and 90Β°. For v = βˆ’3i βˆ’ 4j the calculator still says 53.1Β°, but the true direction is 53.1 + 180 = 233.1Β°. Always sketch.

Calculate

Your turn β€” direction

4The vector v = 3i + 4j makes an angle ΞΈ with the positive x-axis. Find ΞΈ in degrees to 1 decimal place.
Set-up

tan ΞΈ = 4/3 = 1.3333

ΞΈ = arctan(1.3333) = 53.130…° = 53.1Β° (1 d.p.), in the first quadrant.

Β°
Hint: tan ΞΈ = 4/3.
Sort it

Sort by magnitude

Tap a vector, then its magnitude. Square the components and add β€” no calculator needed.

5️⃣ Magnitude 5

7️⃣ Magnitude 7

1️⃣3️⃣ Magnitude 13

Vectors Β· position

Position vectors and distance

The position vector of A is OA = a, measured from the origin. Then:

AB = b βˆ’ a“destination minus start” β€” the single most useful line in the topic

The distance between two points in 3D is the magnitude of that vector:

d = √[(xβ‚‚βˆ’x₁)Β² + (yβ‚‚βˆ’y₁)Β² + (zβ‚‚βˆ’z₁)Β²]
Worked example β€” A(1, 2, 3) and B(4, 6, 15)

AB = b βˆ’ a = (4 βˆ’ 1)i + (6 βˆ’ 2)j + (15 βˆ’ 3)k = 3i + 4j + 12k.

|AB| = √(9 + 16 + 144) = √169 = 13.

Exam trap: AB = b βˆ’ a, not a βˆ’ b. Getting it backwards gives a vector of the right length pointing the wrong way β€” fine for a distance, fatal for a direction or a ratio.

Calculate

Your turn β€” distance in 3D

5A is the point (1, 2, 3) and B is (4, 6, 15). Find the exact distance AB.
Set-up

AB = b βˆ’ a = 3i + 4j + 12k.

|AB| = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

Hint: AB = 3i + 4j + 12k. Now find its magnitude.
Quick check

Position vectors

?OA = a and OB = b. What is the vector AB?
Vectors Β· geometry

Ratios, midpoints and collinearity

Vectors turn geometry into algebra.

  • Midpoint of AB has position vector Β½(a + b).
  • The point P dividing AB in the ratio m : n has OP = a + [m/(m + n)](b βˆ’ a).
  • Collinear points: A, B, C are collinear if AB is a scalar multiple of BC (parallel and sharing the point B).
Worked example β€” A(1, βˆ’2, 4), B(5, 2, βˆ’4), and P divides AB in the ratio 1 : 3

AB = b βˆ’ a = 4i + 4j βˆ’ 8k.

OP = a + ΒΌ(AB) = (1 + 1)i + (βˆ’2 + 1)j + (4 βˆ’ 2)k = 2i βˆ’ j + 2k, so P is (2, βˆ’1, 2).

|OP| = √(4 + 1 + 4) = √9 = 3.

Exam trap: the ratio 1 : 3 means P is one quarter of the way from A to B (1 part out of 1 + 3 = 4), not one third. Count the total parts.

Calculate

Your turn β€” ratio and magnitude

6A is (1, βˆ’2, 4) and B is (5, 2, βˆ’4). The point P divides AB in the ratio 1 : 3. Find the distance OP from the origin.
Set-up

b βˆ’ a = 4i + 4j βˆ’ 8k, so ΒΌ(b βˆ’ a) = i + j βˆ’ 2k.

OP = (i βˆ’ 2j + 4k) + (i + j βˆ’ 2k) = 2i βˆ’ j + 2k, so P = (2, βˆ’1, 2).

|OP| = √(4 + 1 + 4) = √9 = 3.

Hint: OP = a + ΒΌ(b βˆ’ a). Find P first, then its magnitude.
Quick check

Collinear points

?How do you show that the points A, B and C are collinear?
Match it

Vector flashcards

Tap a card on the left, then its meaning.

Object
Meaning
Vectors Β· in context

Using vectors in context

Vectors are the language of mechanics (Paper 3): forces, velocities and displacements all add as vectors.

Worked example β€” resultant force

Two forces act on a particle: F₁ = (3i + 4j) N and Fβ‚‚ = (βˆ’i + 2j) N.

Resultant R = F₁ + Fβ‚‚ = 2i + 6j N.

|R| = √(4 + 36) = √40 = 6.32 N, at arctan(6/2) = 71.6° from the i-direction.

  • A particle in equilibrium has resultant force 0, so the i-components sum to zero and the j-components sum to zero β€” two equations.
  • s = ut + Β½atΒ² works exactly the same way with vector u and a.

Not on 9MA0: the scalar (dot) product and the vector equation of a line belong to Further Mathematics. If a question seems to need the angle between two general vectors, you are looking at the wrong specification.

Recap

The big ideas to take away

Notation: a = xi + yj + zk in 3D; |a| = √(x² + y² + z²)

Unit vector: Γ’ = a/|a| β€” magnitude 1

Parallel: b = Ξ»a Β· Collinear: AB = Ξ»BC through a shared point

Position vectors: AB = b βˆ’ a; midpoint Β½(a + b); ratio m : n β‡’ a + m/(m+n) (b βˆ’ a)

Distance in 3D: √[(xβ‚‚βˆ’x₁)Β² + (yβ‚‚βˆ’y₁)Β² + (zβ‚‚βˆ’z₁)Β²]

In context: forces and velocities add as vectors; equilibrium β‡’ each component sums to 0

Topic 10 is short but heavily examined β€” and it is the bridge into the Mechanics paper. Press Finish to see your score.

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