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Vectors

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Cambridge International AS & A Level Mathematics 9709 — Pure Mathematics 3

Vectors

The big idea: A vector line equation r = a + tb is a starting point plus any multiple of a direction. Questions about lines, intersections, angles and perpendiculars all come down to equating components or using the scalar product.

Summary — What This Chapter Covers

Vectors can be written as column vectors or with i, j, k. AB = b − a.
|xi + yj + zk| = √(x² + y² + z²); a unit vector is a vector divided by its magnitude.
The midpoint of AB has position vector ½(a + b).
Line: r = a + tb, where a is the position vector of a point on the line and b is a direction vector.
Two lines in 3D are parallel, intersect, or are skew.
a·b = |a||b|cos θ = x₁x₂ + y₁y₂ + z₁z₂; perpendicular ⇔ a·b = 0.

1. Vector Operations and Magnitude

Basics

Position, displacement and unit vectors

Adding vectors follows one path after another; multiplying by a scalar k stretches the vector by k (reversing it if k < 0), so parallel vectors are scalar multiples of each other. A position vector OA starts at the origin; a displacement vector AB = OB − OA joins two points.

“OABC is a parallelogram” means OB = OA + OC, and opposite sides are equal vectors: AB = OC.

Results AB = b − a |AB| = √(x² + y² + z²) unit vector = v/|v| midpoint ½(a + b)

Worked Example

A is (1, 2, −1) and B is (3, −1, 5). Find AB, |AB|, a unit vector in the direction of AB, and the midpoint of AB.

Practice Question

ABCD is a parallelogram with A(1, 0, 2), B(4, 1, 3) and C(6, 5, 1). Find the position vector of D.

2. Equations of Lines

Lines

r = a + tb, and how two lines meet

In r = a + tb, a is the position vector of a known point on the line, b is any direction vector, and t is a scalar parameter: each value of t gives one point. A line through A and B is r = a + t(b − a).

To compare two lines: if their directions are scalar multiples, they are parallel. Otherwise equate the components to get three equations in s and t; solve two of them and test the third. If it holds, the lines intersect; if not, they are skew.

Line through A and B r = a + t(b − a)

Worked Example

Lines l: r = (i + 2k) + s(i − j + 2k) and m: r = (3i + j + k) + t(−i − 2j + 3k). Show that l and m intersect and find the point of intersection.

Practice Question

Show that the line l above and the line n: r = (j + k) + u(i + j + k) are skew.

3. Scalar Product and Angles

Scalar product

Angles and perpendiculars

Memorise

Displacement

AB = b − a; midpoint ½(a + b)

Line

r = a + tb: a point plus a multiple of a direction

Angle

cos θ = a·b/(|a||b|) using direction vectors

Perpendicular

a·b = 0; foot of perpendicular from PF·b = 0

Common Mistakes

Same parameter for both lines

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Vectors

Vectors

Position, displacement and unit vectors

r = a + tb, and how two lines meet

Angles and perpendiculars

Displacement

Line

Angle

Perpendicular

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