Question

If the direction vectors of two straight lines are given by \( \vec{m} = \hat{i} + \hat{j} + 0\hat{k} \) and \( \vec{n} = \hat{i} + 0\hat{j} + \hat{k} \), calculate the value of \( \cos\phi \) representing the acute angle between them.

• A. \( \frac{1}{2} \)
• B. \( \frac{\sqrt{3}}{2} \)
• C. \( 0 \)
• D. \( 1 \)

Hint:

Always use absolute value bars for the numerator when calculating the angle between lines. This ensures your cosine value remains positive, which guarantees an acute angle result (\( \le 90^\circ \)) in line with standard geometric conventions.

Answer 1

The Correct Option is A

Concept: The cosine of the angle \( \phi \) between two intersecting lines depends on the scalar dot product of their direction vectors divided by the product of their individual vector magnitudes: \[ \cos\phi = \frac{|\vec{m} \cdot \vec{n}|}{|\vec{m}||\vec{n}|} \]

Step 1: Calculate the scalar dot product of the direction vectors.
Multiply corresponding vector components together and sum the results: \[ \vec{m} \cdot \vec{n} = (1)(1) + (1)(0) + (0)(1) = 1 + 0 + 0 = 1 \]

Step 2: Calculate the individual magnitudes of both vectors.
1. Magnitude of vector \( \vec{m} \): \[ |\vec{m}| = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{1 + 1} = \sqrt{2} \] 2. Magnitude of vector \( \vec{n} \): \[ |\vec{n}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

Step 3: Substitute these values into the angle equation.
Plug the calculated values back into the directional cosine template: \[ \cos\phi = \frac{1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2} \]