MTH1020 · Analysis of Change
Vectors: Dot Product & Projections
Week 3 of Monash MTH1020 Analysis of Change introduces vectors: quantities with both magnitude and direction, their component form in ℝ² and ℝ³, and the arithmetic of addition, subtraction and scalar multiplication. Its examinable core is the dot (scalar) product — computed two ways, a · b = |a||b|cos θ and a · b = a₁b₁ + a₂b₂ + a₃b₃ — which delivers the angle between vectors, a perpendicularity test, and scalar and vector projections. These calculations appear in the mid-semester tests and the final, where correct components, magnitudes and units earn the marks.
What this chapter covers
- 01Vectors vs scalars; equal vectors; the zero vector; representation as arrows (head and tail)
- 02Component form v = a i + b j + c k (or (a, b, c)); position vectors; AB = (position of B) − (position of A)
- 03Vector arithmetic: componentwise addition/subtraction, scalar multiplication, the parallelogram law
- 04Magnitude |v| = √(a²+b²+c²) (Pythagoras) and the unit vector v̂ = v/|v|
- 05Dot product two ways: a · b = |a||b|cos θ and a · b = a₁b₁ + a₂b₂ + a₃b₃; v · v = |v|²
- 06Angle between vectors: cos θ = (a · b)/(|a||b|); perpendicularity test a ⊥ b ⇔ a · b = 0
- 07Scalar projection comp_a b = (a · b)/|a| (signed length of b along a)
- 08Vector projection proj_a b = ((a · b)/|a|²) a; physical motivation via work = F · displacement
Angle and projections via the dot product
- +1Dot product (component form). a · b = (2)(1) + (−1)(3) + (2)(−1) = 2 − 3 − 2 = −3.
- +1Angle between the vectors. First |a| = √(2² + (−1)² + 2²) = √(4 + 1 + 4) = √9 = 3 and |b| = √(1² + 3² + (−1)²) = √(1 + 9 + 1) = √11. Then cos θ = (a · b)/(|a||b|) = −3/(3√11) = −1/√11 ≈ −0.302, so θ = arccos(−1/√11) ≈ 107.5°.
- +1Scalar projection of b onto a. comp_a b = (a · b)/|a| = −3/3 = −1. The negative sign says the component of b along a points opposite to a.
- +1Vector projection of b onto a. proj_a b = ((a · b)/|a|²) a = (−3/9)(2, −1, 2) = (−1/3)(2, −1, 2) = (−2/3, 1/3, −2/3).
Key terms
- Dot (scalar) product
- a · b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃, a scalar. It equals |v|² when a = b = v and underlies the angle formula and perpendicularity test.
- Magnitude |v|
- The length of a vector, |v| = √(a²+b²+c²) by Pythagoras for v = (a, b, c). A unit vector has magnitude 1.
- Unit vector (v̂)
- The vector of length 1 in the direction of a non-zero v: v̂ = v/|v|. Used to specify a direction independent of magnitude.
- Angle between vectors
- Given by cos θ = (a · b)/(|a||b|). A positive dot product gives an acute angle, negative an obtuse one, zero a right angle.
- Scalar projection (comp_a b)
- The signed length of b's component along a: comp_a b = (a · b)/|a|. Negative when that component points opposite to a.
- Vector projection (proj_a b)
- The vector component of b along a: proj_a b = ((a · b)/|a|²) a = (comp_a b) â. It is a vector, not a length.
Vectors: Dot Product & Projections FAQ
What's the difference between the scalar and vector projection?
The scalar projection comp_a b = (a · b)/|a| is a single signed number — the length of b's shadow along a, negative if that shadow points backwards. The vector projection proj_a b = ((a · b)/|a|²) a is that shadow as an actual vector pointing along a. The link is proj_a b = (comp_a b) â, where â is the unit vector along a. Watch the denominators: divide by |a| for the scalar version and by |a|² for the vector version.
How do I know if two vectors are perpendicular?
Two non-zero vectors are perpendicular exactly when their dot product is zero: a ⊥ b ⇔ a · b = 0. This follows from a · b = |a||b|cos θ, since cos 90° = 0. It is the fastest examinable test and reappears constantly in the Week-4 work on normals to planes.
Which form of the dot product should I use?
Use the component form a · b = a₁b₁ + a₂b₂ + a₃b₃ to compute a number from coordinates, and the geometric form a · b = |a||b|cos θ when angles are involved. The two are equal, so you often combine them: compute the dot product and the magnitudes from components, then solve cos θ = (a · b)/(|a||b|) for the angle. Keep units consistent and don't forget the square root in each magnitude.
Can AI help me with vector problems in MTH1020?
Yes. Sia can check your component arithmetic, walk through cos θ = (a · b)/(|a||b|) one line at a time, and show why the scalar and vector projections use different powers of |a| — step by step on your own practice questions. It explains the method and checks your reasoning; it does not do graded assessment for you, and Monash academic-integrity rules apply.
Exam move
Make the dot product automatic, because everything in Week 3 flows from it. Drill the pattern 'compute a · b, compute |a| and |b|, then answer whatever is asked' — angle via cos θ = (a · b)/(|a||b|), perpendicularity via a · b = 0, scalar projection via /|a|, vector projection via /|a|². Keep a worked card that contrasts the two projections side by side so you never divide by the wrong power. Sanity-check with signs: a negative dot product must give an obtuse angle and a negative scalar projection. This material is examined in the mid-semester tests and again in the final, and it is the foundation for the Week-4 cross product, planes and distances, so over-learn it now. Ask Sia to generate fresh vector triples and check your working step by step.
Working through Vectors: Dot Product & Projections in MTH1020? Sia is AskSia’s AI Mathematics tutor — ask any MTH1020 Vectors: Dot Product & Projections question and get a clear, step-by-step explanation grounded in how MTH1020 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
