Question

If \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors such that \( 2\mathbf{a} + \mathbf{b} = 3 \), then which of the following statement is true?

• A. \( \mathbf{a} \) is parallel to \( \mathbf{b} \)
• B. \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \)
• C. \( \mathbf{a} \) is perpendicular to \( 2\mathbf{a} + \mathbf{b} \)
• D. \( \mathbf{b} \) is parallel to \( 2\mathbf{a} + \mathbf{b} \)

Hint:

If vectors are parallel, their direction cosines are the same, and any linear combination of them will result in a vector with a direction that is a scalar multiple of the original vectors.

Answer 1

The Correct Option is A

We are given that \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, and the equation is: \[ 2\mathbf{a} + \mathbf{b} = 3 \] Since \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, we know that: \[ |\mathbf{a}| = 1 \text{and} |\mathbf{b}| = 1 \] Step 1: Check if \( \mathbf{a} \) is parallel to \( \mathbf{b} \) To determine if \( \mathbf{a} \) is parallel to \( \mathbf{b} \), we examine the given equation. For the left-hand side \( 2\mathbf{a} + \mathbf{b} \) to equal 3, the vectors must point in the same direction, which suggests that \( \mathbf{a} \) and \( \mathbf{b} \) must be parallel. If they were not parallel, the sum of the vectors would result in a different magnitude. 

Thus, the equation implies that \( \mathbf{a} \) and \( \mathbf{b} \) are in the same direction. Therefore, \( \mathbf{a} \) is parallel to \( \mathbf{b} \). 

Conclusion: The correct answer is \( \boxed{(a) \, \mathbf{a} \text{ is parallel to } \mathbf{b}} \).