Question

Position vector of \( P \) and \( Q \) are \( \hat{i} + 3\hat{j} - 7\hat{k} \) and \( 5\hat{i} - 2\hat{j} + 4\hat{k} \) respectively. Then the cosine of the angle between \( \overrightarrow{PQ} \) and the y-axis is

• A. \( \frac{4}{\sqrt{162}} \)
• B. \( \frac{5}{\sqrt{162}} \)
• C. \( \frac{5}{\sqrt{162}} \)
• D. \( \frac{4}{\sqrt{162}} \)

Hint:

When finding the cosine of the angle between two vectors, remember to use the formula \( \cos \theta = \frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|\overrightarrow{A}| |\overrightarrow{B}|} \).

Answer 1

The Correct Option is C

Step 1: Find the position vector of \( \overrightarrow{PQ} \).
The position vectors of \( P \) and \( Q \) are:
\[ \overrightarrow{P} = \hat{i} + 3\hat{j} - 7\hat{k}, \quad \overrightarrow{Q} = 5\hat{i} - 2\hat{j} + 4\hat{k} \]
The position vector \( \overrightarrow{PQ} \) is given by: \[ \overrightarrow{PQ} = \overrightarrow{Q} - \overrightarrow{P} = (5\hat{i} - 2\hat{j} + 4\hat{k}) - (\hat{i} + 3\hat{j} - 7\hat{k}) \]
Simplifying: \[ \overrightarrow{PQ} = (5 - 1)\hat{i} + (-2 - 3)\hat{j} + (4 + 7)\hat{k} = 4\hat{i} - 5\hat{j} + 11\hat{k} \]

Step 2: Find the direction vector of the y-axis.
The direction vector of the y-axis is:
\[ \overrightarrow{y} = \hat{j} \]

Step 3: Use the formula for the cosine of the angle between two vectors.
The cosine of the angle \( \theta \) between two vectors \( \overrightarrow{A} \) and \( \overrightarrow{B} \) is given by:
\[ \cos \theta = \frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|\overrightarrow{A}| |\overrightarrow{B}|} \]
Here, \( \overrightarrow{A} = \overrightarrow{PQ} = 4\hat{i} - 5\hat{j} + 11\hat{k} \) and \( \overrightarrow{B} = \overrightarrow{y} = \hat{j} \).

Step 4: Calculate the dot product \( \overrightarrow{A} \cdot \overrightarrow{B} \).
The dot product \( \overrightarrow{A} \cdot \overrightarrow{B} \) is: \[ \overrightarrow{A} \cdot \overrightarrow{B} = (4\hat{i} - 5\hat{j} + 11\hat{k}) \cdot \hat{j} = 4(0) + (-5)(1) + 11(0) = -5 \]

Step 5: Find the magnitudes of \( \overrightarrow{A} \) and \( \overrightarrow{B} \).
The magnitude of \( \overrightarrow{A} = \overrightarrow{PQ} \) is:
\[ |\overrightarrow{A}| = \sqrt{4^2 + (-5)^2 + 11^2} = \sqrt{16 + 25 + 121} = \sqrt{162} \]
The magnitude of \( \overrightarrow{B} = \overrightarrow{y} \) is:
\[ |\overrightarrow{B}| = |\hat{j}| = 1 \]

Step 6: Calculate the cosine of the angle.
Substitute the values into the cosine formula:
\[ \cos \theta = \frac{-5}{\sqrt{162} \times 1} = \frac{-5}{\sqrt{162}} \]
Final Answer.
Thus, the cosine of the angle between \( \overrightarrow{PQ} \) and the y-axis is \( \frac{5}{\sqrt{162}} \), which corresponds to option (C).