Edexcel IAL - Further Pure Mathematics 3- 5.1 Vector and Scalar Triple Products- Study notes  - New syllabus

The Vector Product and the Triple Scalar Product

This topic introduces the vector (cross) product and the triple scalar product, together with their important geometrical interpretations as area and volume. These ideas are fundamental in vector geometry and mechanics.

1. The Vector Product \( \mathbf{a} \times \mathbf{b} \)

The vector product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is a vector:

Perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \)

With direction given by the right-hand rule

Its magnitude is defined by

\( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta \)

where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

Vector Product in Component Form

If

\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \)

\( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \)

then

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \)

Geometrical Interpretation: Area

The magnitude of the vector product represents an area:

Area of the parallelogram with sides \( \mathbf{a} \) and \( \mathbf{b} \):

\( |\mathbf{a} \times \mathbf{b}| \)

Area of the triangle with sides \( \mathbf{a} \) and \( \mathbf{b} \):

\( \dfrac{1}{2}|\mathbf{a} \times \mathbf{b}| \)

2. The Triple Scalar Product

The triple scalar product of vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is defined as

\( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)

It is a scalar quantity.

In component form:

\( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \)

Geometrical Interpretation: Volume

The absolute value of the triple scalar product represents a volume:

Volume of the parallelepiped formed by \( \mathbf{a}, \mathbf{b}, \mathbf{c} \):

\( |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \)

If this value is zero, the vectors are coplanar.