Question

Assertion: If dot product and cross product of $\vec{A}$ and $\vec{B}$ are zero, it implies that one of the vector $\vec{A}$ and $\vec{B}$ must be a null vector.
Reason: Null vector is a vector with zero magnitude.
(A) Both Assertion and Reason are true but Reason is the correct explanation for assertion.
(B) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
(C) Assertion is true and Reason is False.
(D) Assertion is false and Reason is true.

Answer 1

Hint: A vector is a variable that contains magnitude as well as direction. Some mathematical operations, such as multiplication and addition, may be done on vectors. Vector multiplication may be accomplished in two ways: cross product and dot product.



Complete step by step solution:
The resultant of two vectors' scalar products/dot products is always a scalar value. Let’s assume two vectors a and b and the scalar product is computed by multiplying the magnitudes of a, b, and the cosine of the angle between such vectors.
$\vec A \times \vec B$ Denotes a cross product of two vectors $\vec A$and$\vec B$, and the resulting vector is perpendicular to the vectors $\vec A$and$\vec B$.
Dot product of vector $\vec A$ and $\vec B$ can be written as, $\vec A \cdot \vec B = \left| A \right|\left| B \right|\cos \theta = 0$
Cross product of vector $\vec A$ and $\vec B$ can be written as,
$\vec A \times \vec B = \left| A \right|\left| B \right|\sin \theta = 0$
If $\vec A$ and $\vec B$ are not null vectors, then $\sin $ and $\cos $ must both be zero at the same time. However, this is not feasible; hence one of the vectors must be a null vector.
Therefore, correct option is B






Note:
 For tackling these questions we need to have a clear understanding of the concept of vector and scalar quantities. Apart from this we need to have concept clarity of cross and dot product of vectors and their formulas.