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Vectors for SQA National 5 Maths

This page covers the following topics:

1. Vector magnitude

The magnitude of a vector can be calculated using Pythagorasโ€™ theorem. A vector given by (x y) has magnitude โˆš(xยฒ + yยฒ), and a vector given by (x y z) has magnitude โˆš(xยฒ + yยฒ + zยฒ). To obtain the distance between two points, the vector between them can be found and then its magnitude can be calculated to give the required distance.

Vector magnitude

1

Find the distance between the two plotted points.

AB = (12 โˆ’ 4 5 โˆ’ 9) = (8 โˆ’4)
The distance can be calculated using Pythagorasโ€™ theorem.
distance = โˆš(8ยฒ + (โˆ’4)ยฒ) = 4โˆš5

4โˆš5

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2

Find the value of x, given that the distance between (1, 3, 7) and (x, 2, 1) is โˆš53.

Let (1, 3, 7) be A and (x, 2, 1) be B.
AB = (x โˆ’ 1 2 โˆ’ 3 1 โˆ’ 7) = ((x โˆ’ 1) โˆ’1 โˆ’6)

The distance can be calculated using Pythagorasโ€™ theorem.
โˆš53 = โˆš((x โˆ’ 1)ยฒ + (โˆ’1)ยฒ + (โˆ’6)ยฒ)
53 = (x โˆ’ 1)ยฒ + 1 + 36
(x โˆ’ 1)ยฒ = 16
x โˆ’ 1 = 4
x = 4 + 1 = 5

5

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3

Nina has drawn points (1, 4) and (4, 7) on a coordinate axis. Find the distance between the two points.

vector between the points = (4 โˆ’ 1 7 โˆ’ 4) = (3 3)
The distance can be calculated using Pythagorasโ€™ theorem.
distance = โˆš(3ยฒ + (โˆ’3)ยฒ) = 3โˆš2

3โˆš2

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