Correct Answer - Option 2 : Both Statement (I) and Statement (II) are individually true but Statement (II) is not the correct explanation of Statement (I)
Concept:
Zero vector/Null vector:
- The vector having magnitude equal to zero is called a null vector. It is generally represented by O.
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In zero vector the initial and terminal points coincide with each other.
- A point is generally taken as a null vector.
- \(\left| {\vec A} \right| = 0\)
∴ Statement II is correct.
The dot product or scaler of two vectors is given as:
\(D = \overrightarrow{A}.\overrightarrow{B} = \left |\overrightarrow{A} \right |.\left | \overrightarrow{B} \right |~cosθ \)
The cross product or the vector product of two vectors is given as:
\(\overrightarrow{C} = \overrightarrow{A} \times \overrightarrow{B}\)
The magnitude is given as:
\(\left | \overrightarrow{C} \right | = \left |\overrightarrow{A} \right |.\left | \overrightarrow{B} \right |sinθ \)
Analysis:
Given:
\(\vec A \cdot \vec B = \left| {\vec A} \right|\left| {\vec B} \right|\cos \theta = 0\)
\(\vec A \times \vec B = \left| {\vec A} \right|\left| {\vec B} \right|\sin \theta = 0\)
If \(\left. {\overrightarrow {|A} } \right|\)and \(\left. {\overrightarrow {|B} } \right|\)are not null vectors then it implies that sin θ and cos θ both should be zero simultaneously. But this is not possible for any value of θ. ∴ It is essential that one of the vectors must be a null vector. ∴ Both the statements are correct, but Statement II is not a correct explanation of Statement I.
