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Scalar product for SQA Higher Maths

Scalar product

This page covers the following topics:

1. Introduction to scalar product
2. Scalar product and Cartesian coordinates
3. Angle between two 2D vectors
4. Angle between two 3D vectors

The scalar product of two vectors a and b, denoted a ยท b, is defined as | a | | b | cos ฮธ.

Introduction to scalar product

The scalar product of two vectors a = xi + yj + zk and b = pi + qj + wk, denoted a ยท b, is defined as xp + yq + zw.

Scalar product and Cartesian coordinates

If two vectors a and b intersect at an angle of ฮธ, then cosฮธ = a ยท b/| a | | b | where a ยท b is the scalar product of vectors a and b.

Angle between two 2D vectors

If two vectors a and b intersect at an angle of ฮธ, then cosฮธ = a ยท b/|a| |b| where a ยท b is the scalar product of vectors a and b.

Angle between two 3D vectors

1

What is the angle between vectors OA and OB where the coordinates of point A is (3, 1) and B is (2, โˆ’2) with the origin O? Give your answer in radians to 2 decimal places..

OA ยท OB = 3 ร— 2 โˆ’2 = 4, | OA | = โˆš(3ยฒ + 1ยฒ) = โˆš10, | OB | = โˆš( 2ยฒ + (โˆ’2)ยฒ ) = โˆš8. So cosฮธ = 4/( โˆš10 ร— โˆš8 ) which leads to ฮธ = 1.11 radians.

What is the angle between vectors OA and OB where the coordinates of point A is (3, 1) and B is (2, โˆ’2) with the origin O? Give your answer in radians to 2 decimal places..

2

There is an unit cube ABCDโˆ’Aโ‚Bโ‚Cโ‚Dโ‚, vector AB = a, AD = b, AAโ‚ = c, find the angle ฮธ between the straight lines Aโ‚B and AC.

| a = | b | = | c | = 1, a ยท b = b ยท c = c ยท a = 0 since cos90ยบ = 0. Aโ‚B = a โˆ’ c, AC = a + b, so Aโ‚BยทAC = (a โˆ’ c)ยท(a + b) = | a |ยฒ + a ยท b โˆ’ a ยท c โˆ’ b ยท c = 1. On the other hand, | Aโ‚B | = | AC | = โˆš2, so cosฮธ = 1/(โˆš2ยทโˆš2) = 1/โˆš2, which leads to ฮธ = 60ยบ.

There is an unit cube ABCDโˆ’Aโ‚Bโ‚Cโ‚Dโ‚, vector AB = a, AD = b, AAโ‚ = c, find the angle ฮธ between the straight lines Aโ‚B and AC.

3

Three points A, B, C have coordinates (1, 5), (3, 7), (2, 3), find the angle between vectors AB and AC. Give your answer in radians to 2 decimal places.

AB = (3 โˆ’ 1)i + (7 โˆ’ 5)j = 2i + 2j, AC = (2 โˆ’ 1)i + (3 โˆ’ 5)j = i โˆ’ 2j, AB ยท AC = 2 โˆ’ 4 = โˆ’2. So cosฮธ = AB ยท AC/| AB | | AC | = โˆ’2/(โˆš8 ร— โˆš5) = โˆ’โˆš10/10, which leads to ฮธ = 1.89 radians.

Three points A, B, C have coordinates (1, 5), (3, 7), (2, 3), find the angle between vectors AB and AC. Give your answer in radians to 2 decimal places.

4

| a | = 1, | b | = 3 and a + b = (โˆš3, 1). What is the angle between a + b and a โˆ’ b?

cosฮธ = ( (a + b) ยท (a โˆ’ b) )/( | (a + b) | ยท | (a โˆ’ b) | ) = โˆ’1/2. The ฮธ = 2/3ฯ€.

| a | = 1, | b | = 3 and a + b = (โˆš3, 1). What is the angle between a + b and a โˆ’ b?

5

Vectors a = 6i + j โˆ’ 4k, b = 3i โˆ’ 2j, calculate a ยท b.

a ยท b = (6 ร— 3) โˆ’2 = 16.

Vectors a = 6i + j โˆ’ 4k, b = 3i โˆ’ 2j, calculate a ยท b.

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