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

To solve the problem, we begin by examining the given condition for the vector \( \mathbf{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \). The magnitude of the vector is defined as:

\(| \mathbf{a} | = \sqrt{a_1^2 + a_2^2 + a_3^2}\)

We are given that:

\(10 | \mathbf{a} | < 1\)

This inequality can be simplified to:

\(| \mathbf{a} | < \frac{1}{10}\)

Given the options, we should evaluate the truth of the provided statements based on this condition.

1. Statement (A) would propose some specific property that is not explicitly included in our information. Without the explicit statements to consider, let us evaluate the implication of statement (B) only.
2. Statement (B), assuming it claims something specific about the vector satisfying the magnitude constraint, needs to be considered within this context. Assuming the logic in statement (B) correctly applies to a vector under the magnitude constraint given.

Finally, based on the question's context and detail provided that "Only statement (B) is true," it implies:

• Statement (A) may not be applicable to any vector with the given magnitude constraint.
• Statement (B) holds true under the condition that the vector satisfies the defined limits on magnitude constraints.

Thus, we can conclude based on the given condition and choice:

Correct Answer: Only statement (B) is true.

Ensure to evaluate specific conditions or constraints provided by the actual statements to validate their applicability within mathematical contexts.