## Question 5: A and B are two non-zero vectors. (a) How can their scalar product be zero? And (b) how can their vector product be zero?

Two non-zero vectors A and B are given. We have to show how their scalar and vector product can be zero.

(a) Scalar Product

Scalar product of the given vectors can be defined as,

This means the scalar product of the vectors depends upon,

• Magnitude of A
• Magnitude of B
• Cosine of the angle θ between them

Thus the scalar product will be zero if any of the above three quantities are zero. Since the magnitudes of A and B are given to be non-zero, therefore,

if the cosine of the angle θ is zero, then the scalar product of the given vectors can be zero. Since cos90° = 0, therefore, if the angle between A and B is 90° then the scalar product of the given vectors is zero.

Vector Product

Vector product of the given vectors can be defined as,

This means the vector product of the given vectors depends upon,

• Magnitude of A
• Magnitude of B
• Sine of the angle θ between them

Thus the vector product will be zero if any of the above three quantities are zero. Since the magnitudes of A and B are given to be non-zero, therefore, if the sine of the angel θ is zero,

then the scalar product of the given vectors can be zero. Since sin0° = 0, therefore, if the angle between A and B is 0° then the vector product of the given vectors is zero.