The scalar product, angles between vectors and components
Week 10 · The scalar product: definition, properties, coordinates, the angle between two vectors, the component and the projection of one vector on another · 3 lessons
3
The component and the projection of one vector on another
Pure Mathematics Sec 3 Term 1, Unit 3, Lesson 3
GoalFind the component of a vector in the direction of another vector, and its projection.
New words
component of A along B · the number A · B ÷ ‖B‖ = ‖A‖ cos θ; positive if the angle is acute, negative if obtuseprojection vector · (A · B ÷ ‖B‖²) B, the part of A that lies along Bfoot of the perpendicular · the point where the perpendicular from the tip of A meets the line of B
Explanation
Imagine light falling straight on the line of B from the side: the shadow of A on that line is the projection of A on B (in Arabic مسقط). Its signed length is the component of A along B: AB = ‖A‖ cos θ = (A · B)/‖B‖. It is a number. It is positive when θ is acute, negative when θ is obtuse, and zero when the vectors are perpendicular. The projection vector itself has this length in the direction of the unit vector B/‖B‖: (A · B/‖B‖²) B. In the picture A = (4, 3) and B = (6, 0): the component is 24/6 = 4 and the foot of the perpendicular is (4, 0). For the axes, A · i = Aₓ, A · j = Aᵧ and A · k = Az: the components of the earlier lessons are special cases. In mechanics, the component of a force along a given direction is the part of the force that acts in that direction.
Worked examples
Find the component and the projection of A = (4, −2, 4) on B = (1, 2, 2). A · B = 4 − 4 + 8 = 8 and ‖B‖ = 3, so the component is 8/3 ≈ 2.67. The projection vector is (8/9)(1, 2, 2) = (8/9, 16/9, 16/9).
A force F = (10, −3, 4) N acts on a point that moves from A(0, 1, 2) to B(2, 4, 8). Find the component of F along AB. AB = (2, 3, 6) with norm 7. F · AB = 20 − 9 + 24 = 35, so the component is 35/7 = 5 N.
Signs and axes. For A = (−3, 4, 12): A · i = −3, A · j = 4, A · k = 12. The component along i is negative: A leans against the positive x-direction. The component of A along (0, 0, 7) is 84/7 = 12, the same as along k.
Class activity
Shadow game. Hold a pencil under a lamp above a sheet of paper so that its shadow falls on a line drawn on the sheet. Make the pencil vertical to the line, then tilt it. When is the shadow longest, and when is it zero?
Practice
1
Find the component of (3, −1, 2) in the direction of (1, −2, 2).
A · B = 3 + 2 + 4 = 9, ‖B‖ = 3, so the component is 3.
2
Find the projection vector of (6, 2, −3) on (2, 1, −2).
A · B = 12 + 2 + 6 = 20 and ‖B‖² = 9, so the projection is (20/9)(2, 1, −2) = (40/9, 20/9, −40/9).
3
Find the component of the plane vector (2, 5) along (−3, 4).
A · B = −6 + 20 = 14 and ‖B‖ = 5, so the component is 14/5 = 2.8.
4
Find the component of (1, −3, 2) along (3, 2, 1) and say what its sign means.
A · B = 3 − 6 + 2 = −1, ‖B‖ = √14, so the component is −1/√14 ≈ −0.27. It is negative: the angle is obtuse, so A leans slightly against the direction of B.
5
A cart runs on a straight rail in the direction (3, 4, 0). A push of (10, 5, 12) N acts on it. How large is the part of the push along the rail?
F · d = 30 + 20 + 0 = 50 and ‖d‖ = 5, so the part along the rail is 10 N.
