If the position vectors of P and Q are `hati+2hatj-7hatk and 5hati-2hatj+4hatk` respectively, the cosine of the angle between PQ and Z-axis is

Options

A. `(4)/(sqrt(162))`
B. `(11)/(sqrt(162))`
C. `(5)/(sqrt(162))`
D. `(-5)/(sqrt(162))`

Explanation

To find the cosine of the angle between the vector PQ and the Z-axis, we will follow these steps:

### Step 1: Identify the position vectors of points P and Q.
The position vectors are given as:
- \( \vec{P} = \hat{i} + 2\hat{j} - 7\hat{k} \)
- \( \vec{Q} = 5\hat{i} - 2\hat{j} + 4\hat{k} \)

### Step 2: Calculate the vector PQ.
The vector \( \vec{PQ} \) is calculated as:
\[ \vec{PQ} = \vec{Q} - \vec{P} \]
Substituting the values:
\[ \vec{PQ} = (5\hat{i} - 2\hat{j} + 4\hat{k}) - (\hat{i} + 2\hat{j} - 7\hat{k}) \]
This simplifies to:
\[ \vec{PQ} = (5 - 1)\hat{i} + (-2 - 2)\hat{j} + (4 + 7)\hat{k} \]
\[ \vec{PQ} = 4\hat{i} - 4\hat{j} + 11\hat{k} \]

### Step 3: Identify the direction vector of the Z-axis.
The direction vector of the Z-axis is:
\[ \vec{Z} = \hat{k} \]

### Step 4: Calculate the dot product \( \vec{PQ} \cdot \vec{Z} \).
The dot product is given by:
\[ \vec{PQ} \cdot \vec{Z} = (4\hat{i} - 4\hat{j} + 11\hat{k}) \cdot \hat{k} \]
Calculating this:
\[ \vec{PQ} \cdot \vec{Z} = 0 + 0 + 11 = 11 \]

### Step 5: Calculate the magnitudes of \( \vec{PQ} \) and \( \vec{Z} \).
The magnitude of \( \vec{PQ} \) is:
\[ |\vec{PQ}| = \sqrt{(4^2) + (-4^2) + (11^2)} = \sqrt{16 + 16 + 121} = \sqrt{153} \]

The magnitude of \( \vec{Z} \) is:
\[ |\vec{Z}| = |\hat{k}| = 1 \]

### Step 6: Calculate the cosine of the angle \( \theta \).
Using the formula for cosine of the angle between two vectors:
\[ \cos \theta = \frac{\vec{PQ} \cdot \vec{Z}}{|\vec{PQ}| |\vec{Z}|} \]
Substituting the values:
\[ \cos \theta = \frac{11}{\sqrt{153} \cdot 1} = \frac{11}{\sqrt{153}} \]

### Final Answer:
The cosine of the angle between vector PQ and the Z-axis is:
\[ \cos \theta = \frac{11}{\sqrt{153}} \]