Vectors (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
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Vectors (AS Unit 1: Pure Mathematics A) explained
In two dimensions a vector has both magnitude and direction, drawn as a directed line segment.
Read the full explanation
It can be written in column form (x; y) or using unit base vectors i = (1; 0) and j = (0; 1), so v = xi + yj. Vectors add and subtract componentwise: (a; b) + (c; d) = (a + c; b + d), and subtraction reverses direction. Scalar multiplication λ(x; y) = (λx; λy) scales magnitude by |λ|, reversing direction if λ < 0. A position vector gives a point's displacement from the origin. The magnitude is |v| = √(x² + y²), and the unit vector in the direction of v is v̂ = v / |v|. Direction is the angle θ from the positive x-axis, with tan θ = y/x, choosing the correct quadrant from the signs of x and y.
Your focus
- Represent two-dimensional vectors using column notation and unit base vectors i and j.
- Add, subtract and multiply vectors by scalars, and use position vectors.
- Calculate the magnitude of a 2D vector and determine its direction angle and unit vector.
Vectors (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- Converting between column vector notation and unit base vector notation xi + yj, e.g. (3; −2) = 3i − 2j.
- Adding and subtracting vectors componentwise, e.g. (2; 5) + (3; −1) = (5; 4), and interpreting subtraction as adding the negative.
- Multiplying a vector by a scalar, e.g. 3(2; −1) = (6; −3), and stating the effect on magnitude and direction.
- Using position vectors to find the vector between two points, e.g. AB = b − a for position vectors a and b.
- Calculating magnitude |v| = √(x² + y²) and the unit vector v̂ = v / |v|, and finding direction using tan θ = y/x with the correct quadrant.
Examiner Tips
- 💡Sketch the vector components on a quick set of axes to identify the correct direction angle and quadrant.
- 💡Remember that unit base vectors i and j must be written as bold type or underlined in handwritten work.
Common Mistakes
- Confusing column vector notation with coordinate pairs or fractions; correction: (x; y) denotes a displacement, not a point, and is written vertically.
- Giving the acute angle from tan θ = |y/x| without adjusting for the correct quadrant or bearing; correction: use the signs of x and y to place the angle in the correct quadrant.
- Adding vectors by adding magnitudes rather than components; correction: vectors add componentwise, so (1; 2) + (3; 4) = (4; 6), not a vector of magnitude 1 + 3.