If the position vectors of P and Q are `hati+2hatj-7hatk` and `5hati-3hatj+4hatk` respectively, the cosine of the angle between `vec(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 \(\vec{PQ}\) and the z-axis, we can follow these steps:

### Step 1: Identify the position vectors
The position vectors of points P and Q are given as:
- \(\vec{P} = \hat{i} + 2\hat{j} - 7\hat{k}\)
- \(\vec{Q} = 5\hat{i} - 3\hat{j} + 4\hat{k}\)

### Step 2: Find the vector \(\vec{PQ}\)
The vector \(\vec{PQ}\) can be found using the formula:
\[ \vec{PQ} = \vec{Q} - \vec{P} \]
Substituting the values:
\[ \vec{PQ} = (5\hat{i} - 3\hat{j} + 4\hat{k}) - (\hat{i} + 2\hat{j} - 7\hat{k}) \]
Now, simplifying this:
\[ \vec{PQ} = (5 - 1)\hat{i} + (-3 - 2)\hat{j} + (4 + 7)\hat{k} \]
\[ \vec{PQ} = 4\hat{i} - 5\hat{j} + 11\hat{k} \]

### Step 3: Find the magnitude of \(\vec{PQ}\)
The magnitude of \(\vec{PQ}\) is given by:
\[ |\vec{PQ}| = \sqrt{(4)^2 + (-5)^2 + (11)^2} \]
Calculating this:
\[ |\vec{PQ}| = \sqrt{16 + 25 + 121} = \sqrt{162} \]

### Step 4: Find the cosine of the angle with the z-axis
The cosine of the angle \(\gamma\) between the vector \(\vec{PQ}\) and the z-axis (represented by the unit vector \(\hat{k}\)) can be calculated using the formula:
\[ \cos \gamma = \frac{\vec{PQ} \cdot \hat{k}}{|\vec{PQ}|} \]
Calculating the dot product \(\vec{PQ} \cdot \hat{k}\):
\[ \vec{PQ} \cdot \hat{k} = (4\hat{i} - 5\hat{j} + 11\hat{k}) \cdot \hat{k} = 11 \]
Now substituting into the cosine formula:
\[ \cos \gamma = \frac{11}{\sqrt{162}} \]

### Final Answer
Thus, the cosine of the angle between \(\vec{PQ}\) and the z-axis is:
\[ \cos \gamma = \frac{11}{\sqrt{162}} \]