Question

\(\vec A\) is a vector quantity such that
\(|\vec A|\) = non-zero constant.
Which of the following expression is true for \(\vec A\)?

• A. \(\vec A.\vec A=0\)
• B. \(\vec A \times \vec A<0\)
• C. \(\vec A \times\vec A=0\)
• D. \(\vec A \times\vec A>0\)

Answer 1

The Correct Option is C

 To determine the correct expression for the vector \(\vec A\), where \(|\vec A|\) is a non-zero constant, let's analyze the given options using vector algebra principles.

• \(\vec A.\vec A=0\): This option implies that the dot product of \(\vec A\) with itself is zero. However, the dot product of a vector with itself is equal to the square of its magnitude, i.e., \(\vec A \cdot \vec A = |\vec A|^2\). Since \(|\vec A|\) is a non-zero constant, its square cannot be zero. Therefore, this option is incorrect.
• \(\vec A \times \vec A < 0\): The cross product of a vector with itself is always zero, as the sine of the angle between any vector and itself is zero since the angle is zero degrees. Hence, \(\vec A \times \vec A\) cannot be negative. This option is incorrect.
• \(\vec A \times \vec A = 0\): The cross product of any vector with itself is zero because the sine of zero degrees (the angle between the vector and itself) is zero. Therefore, \(\vec A \times \vec A = 0\) is true. This is the correct option.
• \(\vec A \times \vec A > 0\): Similarly to the explanation above, the cross product of a vector with itself cannot be greater than zero, as it results in zero. This option is incorrect.

Therefore, the correct expression for the vector \(\vec A\) is \(\vec A \times \vec A = 0\).