Mechanics II – SS2 Further Mathematics Lesson Note (Third Term)

The vector product (cross product) of two vectors a and b is a vector whose magnitude is |a||b| sin θ, where θ is the angle between them. Its direction is perpendicular to both a and b, in the sense of a rotation from a to b. It is written a × b.

Behavioural objectives

By the end of the lesson, students should be able to:

• State the definition of the vector product of two vectors.
• Write the vector product in the form a × b = |a||b| sin θ n̂.
• Explain why the vector product is not commutative.
• State and use the properties of the vector product.

What is the vector product of two vectors?

Given two vectors a and b whose directions are inclined at an angle θ, their vector product is a vector r. The magnitude of r is |a||b| sin θ. The direction of r is perpendicular to both a and b, and is positive relative to a rotation from the vector a to the vector b.

The vector product of a and b is written a × b.

Thus:

r = a × b = |a||b| sin θ n̂

where n̂ is a unit vector perpendicular to the plane of a and b.

What are the properties of the vector product?

The vector product is not commutative

For 0 ≤ θ ≤ π:

b × a = |b||a| sin(−θ) n̂
= −|a||b| sin θ n̂
= −a × b

Thus the vector product of two vectors is not commutative.

Multiplication by a scalar

Where k is a scalar:

(ka) × b = a × (kb) = k(a × b) = k|a||b| sin θ n̂

Distributive law

a × (b + c) = a × b + a × c

Evaluation

1. Define the vector product of two vectors a and b.
2. What is the direction of a × b?
3. Show that b × a = −(a × b).
4. If k is a scalar, write (ka) × b in terms of a × b.

Assignment

1. Given that p = 2i + 3j + 4k and q = 5i − 6j + 7k, find:
   1. p × q
   2. (p + q) . (p − q)
2. Given that a = 4i − 5j + 2k and b = −7i + 3j − 6k:
   1. Find the scalar product of a and b.
   2. Find the direction cosines of 2a + 3b.
3. Find the angle between p = 6i + 2j − 4k and q = 9i + 5j.