What is the magnitude of the cross product \(\overrightarrow{a}\) × \(\overrightarrow{b}\)?

Solution:

The cross product of two vectors is another vector that is perpendicular to both the given vectors.

Physical significance of the cross product of \(\overrightarrow{a}\) × \(\overrightarrow{b}\):

1) The magnitude of a cross product is the area of the parallelogram that they determine.

2) The direction of the cross product is orthogonal (perpendicular) to the plane determined by the two vectors.

Formulation of \(\overrightarrow{a}\) × \(\overrightarrow{b}\)

\(\overrightarrow{a}\) × \(\overrightarrow{b}\) = |a| |b| sinA  (A is the angle formed between the vectors a and b)

Look at two examples shown in the figure below.

a) \(\overrightarrow{a}\) × \(\overrightarrow{b}\) = |a||b|sinA = 6 × 4 × sin 45º = 12√2

b) \(\overrightarrow{a}\) × \(\overrightarrow{b}\) = |a||b|sinA = 6 × 4 × sin 180º = 0

Thus, the magnitude of the cross product of two vectors an and b is the area of the parallelogram formed by a and b, that is, |a|.|b|

What is the magnitude of the cross product vectors a × b?

Summary:

The magnitude of the cross product of two vectors an and b is the area of the parallelogram formed by a and b, that is, |a|.|b|