3D vectors for AQA A-level Maths

3D vectors

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”.

Adding 3D vectors

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

Subtracting 3D vectors

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.

Multiplying 3D vectors

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

Magnitude of 3D vectors

Vectors are perpendicular if cosΘ = 0.

Proof with 3D vectors

1

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

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PR = 14i + 6j + 2k

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

2

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

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3√14

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

3

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

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a = 1 and b = 2

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

4

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

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8i − 2j

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

5

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

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

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

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