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Subjects · Leaving Cert Applied Maths

Leaving Cert Applied Maths: Vectors & the dot product

How often Vectors & the dot product comes up on the Applied Maths papers, every year it was asked, and questions to try.

HL Asked on 2 of the last 3 Higher Level papers, most recently in 2025. most years

OL Asked on 3 of the last 3 Ordinary Level papers, most recently in 2025. banker

Vectors & the dot product, Higher Level(7 marks)

Quick ones on Vectors & the dot product.

(a)Magnitude of the vector 6i + 8j?
  1. 14
  2. 10
  3. 100
(b)Two non-zero vectors are perpendicular exactly when…?
  1. Their dot product is 1
  2. Their magnitudes are equal
  3. Their dot product is 0
(c)(3i + 4j)·(2i − j) = ?
  1. 2
  2. 6i − 4j
  3. 10
Show the answers

(a) 10

(b) Their dot product is 0

(c) 2

Your turn: Higher Level questions on Vectors & the dot product.

Higher Level

Asked on 2 of the last 3 Higher Level papers, most recently in 2025. most years

Every paper, year by year

YearWhere it came up
2025Q5
2024Not asked
2023Q8

Links open the State Examinations Commission’s paper for that year.

More Vectors & the dot product questions

Vectors & the dot product, 3 marks

Find k and t if k(i + 2j) + t(3i − j) = 7i.

  1. k = 2, t = 1
  2. k = 7, t = 0
  3. k = 1, t = 2
Show the answer

k = 1, t = 2

Match parts. i: k + 3t = 7. j: 2k − t = 0, so t = 2k. Then k + 6k = 7 gives k = 1 and t = 2. Check j: 2 − 2 = 0.

Vectors & the dot product, 2 marks

For any vector a, the dot product a·a equals…?

  1. 2|a|
  2. |a|²
  3. 0
Show the answer

|a|²

(xi + yj)·(xi + yj) = x² + y², which is the square of the magnitude. So |a| = √(a·a). The angle between a vector and itself is 0 and cos 0 = 1.

Vectors & the dot product, 3 marks

|a| = 4, |b| = 5 and a·b = 10. Angle between a and b?

  1. 60°
  2. 30°
  3. 90°
Show the answer

60°

cos θ = a·b ÷ (|a||b|) = 10 ÷ 20 = 0.5, so θ = 60°. A dot product of 0 would mean 90°.

Ordinary Level

Asked on 3 of the last 3 Ordinary Level papers, most recently in 2025. banker

Every paper, year by year

YearWhere it came up
2025Q8
2024Q2
2023Q3

Links open the State Examinations Commission’s paper for that year.

More Vectors & the dot product questions

Vectors & the dot product, 3 marks

Find (2i + 3j)·(4i − j).

  1. 11
  2. 8i − 3j
  3. 5
Show the answer

5

Multiply matching parts and add: 2 × 4 + 3 × (−1) = 8 − 3 = 5. A dot product is a number (scalar), not a vector.

Vectors & the dot product, 3 marks

Find k if (ki + 2j)·(3i − 6j) = 0.

  1. k = 2
  2. k = 4
  3. k = −4
Show the answer

k = 4

Dot product: 3k + 2 × (−6) = 3k − 12 = 0, so k = 4. The vectors are then perpendicular.

Vectors & the dot product, 2 marks

Two non-zero vectors are perpendicular when their dot product is…

  1. 0
  2. 1
  3. −1
Show the answer

0

a·b equals the product of the magnitudes times cos θ. At 90°, cos θ = 0, so the dot product is 0.

Other Applied Maths topics

All of Leaving Cert Applied Maths