If vectors `vec(a)=hat(i)-2hat(j)+hat(k), vec(b)=-2hat(i)+4hat(j)+5hat(k)` and `vec(c )=hat(i)-6hat(j)-7hat(k)`, then find the value of `|vec(a)+vec(b)+vec(c )|`.

Answer

To find the value of \(|\vec{a} + \vec{b} + \vec{c}|\), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \]
\[ \vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k} \]
\[ \vec{c} = \hat{i} - 6\hat{j} - 7\hat{k} \]

### Step 2: Add the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\)
We will add the corresponding components of the vectors:

- For the \(\hat{i}\) component:
\[ 1 + (-2) + 1 = 0 \]

- For the \(\hat{j}\) component:
\[ -2 + 4 + (-6) = -4 \]

- For the \(\hat{k}\) component:
\[ 1 + 5 + (-7) = -1 \]

Thus, we have:
\[ \vec{a} + \vec{b} + \vec{c} = 0\hat{i} - 4\hat{j} - 1\hat{k} \]

### Step 3: Write the resultant vector
The resultant vector can be expressed as:
\[ \vec{R} = 0\hat{i} - 4\hat{j} - 1\hat{k} \]

### Step 4: Calculate the magnitude of the resultant vector
The magnitude of a vector \(\vec{R} = x\hat{i} + y\hat{j} + z\hat{k}\) is given by:
\[ |\vec{R}| = \sqrt{x^2 + y^2 + z^2} \]

For our resultant vector:
- \(x = 0\)
- \(y = -4\)
- \(z = -1\)

Calculating the magnitude:
\[ |\vec{R}| = \sqrt{0^2 + (-4)^2 + (-1)^2} = \sqrt{0 + 16 + 1} = \sqrt{17} \]

### Final Answer
Thus, the value of \(|\vec{a} + \vec{b} + \vec{c}|\) is:
\[ \sqrt{17} \]

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