Chapter 9.7, Problem 63AYU

To determine

To explain:about the vector u and v if $\text{u}\cdot \text{v}=0\text{and u}\times \text{v}=0$ .

Explanation of Solution

Given:

The givenvector is $\text{u}\cdot \text{v}=0\text{and u}\times \text{v}=0$ .

Calculation:

Dot product of two vectors i.e. scalar product is a scalar quantity.

Cross product is the vector product of two vectors. 

Cross product has a direction perpendicular to the direction of the direction of both the vectors.

The dot product $\text{u}\cdot \text{v}=0$  will be zero,if u and vare perpendicular (at 90 degrees) to each other.

The dot product $\text{u}\times \text{v}=0$  will be zero,if u and vare parallel to each other.

This means,

If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero.