The scalar product in component form and in terms of magnitudes and the angle between, finding angles, testing perpendicularity and parallelism.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
Multiply two vectors and get a number — a number that tells you the angle between them. If it is zero, the vectors are at right angles.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
See the angle between two vectors and the dot product.
Copy these into your notebook in your own words — that is how they stick.
a · b = x₁x₂ + y₁y₂ = |a||b| cos θ.
Commutative; a · a = |a|²; distributive.
cos θ = a · b / (|a||b|).
a · b = 0.
b = λa; a · b = ±|a||b|.
a · b > 0 acute; = 0 right angle; < 0 obtuse.
Use vectors that both start at that vertex.
Set a · b = 0 for perpendicular, ratio equality for parallel.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (24 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [12 marks]
The points are A(1, 2), B(5, 4) and C(3, −2).
(a) Express AB and AC in the form xi + yj. [2]
(b) Show that angle BAC = 90°. [2]
(c) Calculate angle ABC. [4]
(d) Find the area of triangle ABC and the coordinates of D such that ABDC is a rectangle. [4]
Question 2 [12 marks]
Let a = 2i + kj and b = 3i − 6j.
(a) Find k if a is perpendicular to b. [3]
(b) Find k if a is parallel to b. [3]
(c) Find the acute angle between 3i + j and i + 2j. [4]
(d) Show that p = 5i + 2j and q = −4i + 10j are perpendicular. [2]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.