Question

Statement 1: If dot product and cross product of $\overrightarrow A$ and $\overrightarrow B$ are zero, it implies that one of the vector $\overrightarrow A$ and $\overrightarrow B$ must be a null vector.
Statement 2: Null vector is a vector with zero magnitude.
A) Statement- 1 is false, Statement 2 is true.
B) Statement- 1 is true, Statement 2 is true, Statement 2 is a correct explanation for statement- 1.
C) Statement- 1 is true, statement - 2 is not a correct explanation for statement- 1.
D) Statement 1 is true, statement-2 is false.

Answer 1

The dot product of two perpendicular vectors is zero. Similarly, the cross product of two parallel vectors is zero. No two vectors can be parallel and perpendicular at the same time.

Complete answer:
Vectors are quantities that have magnitude as well as a direction. Normal operations cannot be performed on them. The operations like addition, subtraction, multiplication cannot be performed normally. We do vector addition, vector subtraction and vector multiplication instead of the normal operations.
What vector multiplication?
When two vectors are multiplied, this multiplication is called vector multiplication. However there are two different types of multiplications done.
They are:
1. Cross product or Vector product.
2. Dot product or Scalar product.
The vector product or cross product gives us a vector that is perpendicular to the plane of the two vectors that are multiplied.
We all know that the cross product for any two vectors that are parallel or coinciding or antiparallel is zero.
The scalar product or Dot product gives us the amount of influence of one vector in the direction of the other vector. It is a scalar quantity.
But the scalar product or dot product of two perpendicular vectors is zero.
Now, for the $\vec A$ and $\vec B$ if they are parallel, then their cross product will be zero. But if they are perpendicular then the dot product will be zero. However, it is not possible for two vectors to be perpendicular and parallel at the same time.
Thus the given condition can only be true if the one of the vectors is a null vector. Because a null vector is a vector with zero magnitude.

Therefore the correct option is B.

Note: Result of a dot product is a scalar quantity, the result of a cross product is a vector quantity. Dot product of vectors in the same direction is maximum and minimum when the vectors are in the opposite direction. Cross product follows right hand rule. We can find the direction of the resultant cross product by using right hand rule.