Question

What is the vector product of two parallel vectors?

Answer 1

Hint: We use the concept of parallel vectors and their cross product.
When two vectors are in the same direction and have the same angle but vary in magnitude, it is known as the parallel vector.
The formula is: \[\overrightarrow A \times \overrightarrow B = ||A||B||\operatorname{Sin} \theta n\]

Complete step-by-step solution:
Here we consider two parallel vectors \[\overrightarrow A \] and \[\overrightarrow B \]
We know by a formula that
\[\overrightarrow A \times \overrightarrow B = ||A||B||\operatorname{Sin} \theta n\]
||A|| length of vector A
|| B || length of vector B
\[\theta \] = angle between a and b
\[n\] = unit vector perpendicular to the plane containing and b
If two vectors are parallel then
\[\theta = 0\]Or \[180 = 0\]
Since \[Sin{\text{ }}0{\text{ }} = {\text{ }}sin{\text{ }}180{\text{ }} = {\text{ }}0\]
So putting in the above equation we get \[\overrightarrow A \times \overrightarrow B = 0\]
Hence the vector product of two parallel vectors is equal to zero.
Additional information: Vector product or cross product is a binary operation in three-dimensional geometry. The cross product is used to find the length of a vector or the angle between two vectors. The cross product is used to find a vector perpendicular to the plane spanned by two vectors. It has many applications in both math and physics.

Note: Vector product and cross product can be confusing at times but they mean the same. The concept of parallel vectors is equally important. In this question, it is very important to know the formula for the cross product of two parallel vectors. Using the above formula one can easily simplify the problem and answer the concepts related to three-D geometry used in this question. One should be well versed with the concepts related to three-D geometry.