Question

A unit vector in XY-plane making an angle \(45^\circ\) with \(\hat{i} + \hat{j}\) and an angle \(60^\circ\) with \(3\hat{i} - 4\hat{j}\) is:

• A. \(\frac{13}{14} \hat{i} + \frac{1}{14} \hat{j}\)
• B. \(\frac{1}{14} \hat{i} + \frac{13}{14} \hat{j}\)
• C. \(\frac{13}{14} \hat{i} - \frac{1}{14} \hat{j}\)
• D. \(\frac{1}{14} \hat{i} - \frac{13}{14} \hat{j}\)

Hint:

To find unit vectors at given angles, use trigonometric identities and solve using the dot product.

Answer 1

The Correct Option is A

Step 1: The unit vector \( \mathbf{v} \) in the XY-plane making an angle of \( 45^\circ \) with \( \hat{i} + \hat{j} \) is given by:

\[ \mathbf{v} = \hat{i} \cos(45^\circ) + \hat{j} \sin(45^\circ) \]

Step 2: The angle between \( \mathbf{v} \) and the vector \( 3\hat{i} - 4\hat{j} \) is \( 60^\circ \). Using the dot product formula:

\[ \mathbf{v} \cdot (3\hat{i} - 4\hat{j}) = |\mathbf{v}| \cdot |3\hat{i} - 4\hat{j}| \cdot \cos(60^\circ) \]

Step 3: Solving this system of equations gives the components of the unit vector \( \mathbf{v} \).