Question

For any vector \( \mathbf{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \), with \( 10 | \mathbf{a} |<1 \), \( i = 1, 2, 3 \), consider the following statements:

• A. Only statement (A) is true
• B. Only statement (B) is true
• C. Both (A) and (B) are true
• D. Neither (A) nor (B) is true

Hint:

When dealing with vector magnitude, remember that the magnitude is always the square root of the sum of the squares of its components, which will always be less than or equal to the maximum component value.

Answer 1

The Correct Option is B

Let \( |\mathbf{a}| \) represent the magnitude of vector \( \mathbf{a} \), where: \[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \] We are given the following conditions: \[ \mathbf{A:} \ \max(|a_1|, |a_2|, |a_3|) = |\mathbf{a}| \] \[ \mathbf{B:} \ | \mathbf{a} | \leq \max (|a_1|, |a_2|, |a_3|) \] Now, let's prove each statement.
Statement A: We know that: \[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \] We also have: \[ \max(|a_1|, |a_2|, |a_3|) \geq |a_1|, |a_2|, |a_3| \] Thus, \[ |\mathbf{a}| \leq \max(|a_1|, |a_2|, |a_3|) \] Hence, statement A is false because the magnitude of the vector is less than or equal to the maximum of the absolute values of its components, not necessarily equal.
Statement B: This statement is true because the magnitude of the vector \( \mathbf{a} \) is always less than or equal to the maximum of the absolute values of the components.
Thus, statement (B) is the correct statement.