If `vec(a)=3hat(i)-2hat(j)+hat(k), vec(b)=2hat(i)-4 hat(j)-3 hat(k)`, find `|vec(a)-2 vec(b)|`.

Answer

To solve the problem, we need to find the magnitude of the vector \(\vec{a} - 2\vec{b}\) given the vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + \hat{k}\) and \(\vec{b} = 2\hat{i} - 4\hat{j} - 3\hat{k}\).

### Step 1: Calculate \(2\vec{b}\)
First, we need to calculate \(2\vec{b}\):
\[ 2\vec{b} = 2(2\hat{i} - 4\hat{j} - 3\hat{k}) = 4\hat{i} - 8\hat{j} - 6\hat{k} \]

### Step 2: Calculate \(\vec{a} - 2\vec{b}\)
Now we can find \(\vec{a} - 2\vec{b}\):
\[ \vec{a} - 2\vec{b} = (3\hat{i} - 2\hat{j} + \hat{k}) - (4\hat{i} - 8\hat{j} - 6\hat{k}) \]
Distributing the negative sign:
\[ = 3\hat{i} - 2\hat{j} + \hat{k} - 4\hat{i} + 8\hat{j} + 6\hat{k} \]
Now, combine like terms:
\[ = (3 - 4)\hat{i} + (-2 + 8)\hat{j} + (1 + 6)\hat{k} \]
\[ = -1\hat{i} + 6\hat{j} + 7\hat{k} \]

### Step 3: Find the magnitude of \(\vec{a} - 2\vec{b}\)
The magnitude of a vector \(\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}\) is given by:
\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]
For our vector \(\vec{a} - 2\vec{b} = -1\hat{i} + 6\hat{j} + 7\hat{k}\):
\[ |\vec{a} - 2\vec{b}| = \sqrt{(-1)^2 + (6)^2 + (7)^2} \]
Calculating each term:
\[ = \sqrt{1 + 36 + 49} \]
\[ = \sqrt{86} \]

### Final Answer
Thus, the magnitude \(|\vec{a} - 2\vec{b}|\) is:
\[ \sqrt{86} \]