If vectors A and B be respectively equal to `3hati - 4hatj + 5hatk and 2hati + 3hatj - 4hatk.` Find the unit vector parallel t A + B

Explanation

To find the unit vector parallel to the vector \( \mathbf{A} + \mathbf{B} \), we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \mathbf{A} = 3\hat{i} - 4\hat{j} + 5\hat{k} \]
\[ \mathbf{B} = 2\hat{i} + 3\hat{j} - 4\hat{k} \]

### Step 2: Calculate \( \mathbf{A} + \mathbf{B} \)
We add the two vectors component-wise:
\[ \mathbf{A} + \mathbf{B} = (3\hat{i} + 2\hat{i}) + (-4\hat{j} + 3\hat{j}) + (5\hat{k} - 4\hat{k}) \]
\[ = (3 + 2)\hat{i} + (-4 + 3)\hat{j} + (5 - 4)\hat{k} \]
\[ = 5\hat{i} - \hat{j} + \hat{k} \]

### Step 3: Find the magnitude of \( \mathbf{A} + \mathbf{B} \)
The magnitude of a vector \( \mathbf{V} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:
\[ |\mathbf{V}| = \sqrt{a^2 + b^2 + c^2} \]
For \( \mathbf{A} + \mathbf{B} = 5\hat{i} - \hat{j} + \hat{k} \):
\[ |\mathbf{A} + \mathbf{B}| = \sqrt{(5)^2 + (-1)^2 + (1)^2} \]
\[ = \sqrt{25 + 1 + 1} = \sqrt{27} \]

### Step 4: Find the unit vector parallel to \( \mathbf{A} + \mathbf{B} \)
The unit vector \( \mathbf{u} \) in the direction of \( \mathbf{V} \) is given by:
\[ \mathbf{u} = \frac{\mathbf{V}}{|\mathbf{V}|} \]
Thus, the unit vector parallel to \( \mathbf{A} + \mathbf{B} \) is:
\[ \mathbf{u} = \frac{5\hat{i} - \hat{j} + \hat{k}}{\sqrt{27}} \]

### Final Answer
The unit vector parallel to \( \mathbf{A} + \mathbf{B} \) is:
\[ \mathbf{u} = \frac{5}{\sqrt{27}}\hat{i} - \frac{1}{\sqrt{27}}\hat{j} + \frac{1}{\sqrt{27}}\hat{k} \]

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