# StudySquare

# 3D vectors for AQA A-level Maths

This page covers the following topics:

1. Adding 3D vectors

2. Subtracting 3D vectors

3. Multiplying 3D vectors

4. Magnitude of 3D vectors

5. Proof with 3D vectors

3D vectors are added by adding their corresponding coordinates or by connecting them “head to toe”.

To subtract 3D vectors, reverse the direction of the vector that is subtracted or subtract corresponding coordinates.

Two vectors can be multiplied by using a dot product, which can be obtained by multiplying corresponding coordinates and adding the results up. Alternatively, cosine value of an angle between the vectors may be used.

2D Pythagoras’ theorem can be modified to obtain the magnitude of a 3D vector. The final equation is M = √(a² + b² + c²).

Vectors are perpendicular if cosΘ = 0.

# 1

Find Cartesian vector equation of PR which is connecting points P(−6, 2, −6) and R(8, 8, −4).

PR = 14i + 6j + 2k

# 2

Calculate the magnitude of vector v = 9i + 6j − 3k

3√14

# 3

If u = (0 −4), v = (15 0) and w = (30 −4), find a and b when w = au + bv.

a = 1 and b = 2

# 4

If AB = 3i + j and BC = 5i − 3j, work out an expression for AC.

8i − 2j

# 5

If PR (5, −1, −2) and QR (−5, 1, 2), show that P,Q and R are colinear.

QR and PR have a common point R. QR and PR have a common direction.

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