Vector Projection: HSC Extension Mathematics

Year 12 Extension Mathematics Understanding projection of one vector onto another 30-minute lesson with worked examples

Learning Objectives

Define vector projection and scalar projection Calculate the projection of one vector onto another Apply projection formulas using dot products Solve HSC-style problems involving vector projection Understand geometric interpretation of projections

What is Vector Projection?

Vector projection is the 'overlay' of one vector onto another Creates a new vector in the direction of the target vector Think of it as the 'shadow' cast by one vector onto another Direction matters: projection of a onto b ≠ projection of b onto a

Vector Projection Diagram

Key Notation and Terminology

proj_b(a) = vector projection of a onto b comp_b(a) = scalar projection of a onto b The result is always in the direction of vector b Scalar projection gives magnitude only Vector projection gives both magnitude and direction

Remember

The projection of vector a onto vector b always lies in the direction of vector b

The Scalar Projection Formula

comp_b(a) = |a| cos θ Where θ is the angle between vectors a and b This gives the magnitude of the projection Can be positive or negative depending on angle Uses the dot product: comp_b(a) = (a · b) / |b|

The Vector Projection Formula

proj_b(a) = comp_b(a) × (b/|b|) Or equivalently: proj_b(a) = ((a · b) / |b|²) × b This gives both magnitude and direction Result is always parallel to vector b The formula combines scalar projection with unit vector

Quick Check: Understanding Angles

Consider vectors a and b If θ = 0°, what happens to the projection? If θ = 90°, what happens to the projection? If θ = 180°, what happens to the projection?

Worked Example 1

Given: a = (3, 4) and b = (1, 0) Find the scalar projection of a onto b Step 1: Calculate a · b = 3(1) + 4(0) = 3 Step 2: Calculate |b| = √(1² + 0²) = 1 Step 3: comp_b(a) = (a · b) / |b| = 3/1 = 3

Worked Example 1 (continued)

Now find the vector projection of a onto b Step 1: We know comp_b(a) = 3 Step 2: Find unit vector: b/|b| = (1,0)/1 = (1,0) Step 3: proj_b(a) = comp_b(a) × (b/|b|) = 3 × (1,0) = (3,0) The projection vector is (3,0)

Check Your Understanding

Given vectors a = (2, 6) and b = (3, 4) What is the scalar projection of a onto b? Take 2 minutes to calculate this