If `a=2hati+2hatj-8hatk and b=hati+3hatj-4hatk`, then the magnitude of a+b is equal to

Options

A. `13`
B. `(13)/(5)`
C. `3/13`
D. `4/13`

Explanation

To find the magnitude of the vector \( \mathbf{a} + \mathbf{b} \), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ \mathbf{a} = 2\hat{i} + 2\hat{j} - 8\hat{k} \]
\[ \mathbf{b} = \hat{i} + 3\hat{j} - 4\hat{k} \]

### Step 2: Add the vectors
To find \( \mathbf{a} + \mathbf{b} \), we add the corresponding components:
\[ \mathbf{a} + \mathbf{b} = (2\hat{i} + 2\hat{j} - 8\hat{k}) + (\hat{i} + 3\hat{j} - 4\hat{k}) \]
Combining the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components:
\[ = (2 + 1)\hat{i} + (2 + 3)\hat{j} + (-8 - 4)\hat{k} \]
\[ = 3\hat{i} + 5\hat{j} - 12\hat{k} \]

### Step 3: Find the magnitude of \( \mathbf{a} + \mathbf{b} \)
The magnitude of a vector \( \mathbf{v} = x\hat{i} + y\hat{j} + z\hat{k} \) is given by:
\[ |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \]
For \( \mathbf{a} + \mathbf{b} = 3\hat{i} + 5\hat{j} - 12\hat{k} \):
\[ |\mathbf{a} + \mathbf{b}| = \sqrt{3^2 + 5^2 + (-12)^2} \]
Calculating each term:
\[ = \sqrt{9 + 25 + 144} \]
\[ = \sqrt{178} \]

### Step 4: Final result
Thus, the magnitude of \( \mathbf{a} + \mathbf{b} \) is:
\[ |\mathbf{a} + \mathbf{b}| = \sqrt{178} \]

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