If `'theta'` is the angle between the vectors : `vec(a)=hat(i)+2hat(j)+3hat(k)` and `vec(b)=3hat(i)-2hat(j)+hat(k)`, find sin `theta`.

Answer

To find \(\sin \theta\) where \(\theta\) is the angle between the vectors \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}\), we can follow these steps:

### Step 1: Find the modulus of vector \(\vec{a}\)
The modulus of a vector \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) is given by:
\[ |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
For \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\):
- \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 3\)

Calculating the modulus:
\[ |\vec{a}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 2: Find the modulus of vector \(\vec{b}\)
For \(\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}\):
- \(b_1 = 3\), \(b_2 = -2\), \(b_3 = 1\)

Calculating the modulus:
\[ |\vec{b}| = \sqrt{3^2 + (-2)^2 + 1^2} = \sqrt{9 + 4 + 1} = \sqrt{14} \]

### Step 3: Calculate the cross product \(\vec{a} \times \vec{b}\)
Using the determinant method:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 3 & -2 & 1 \end{vmatrix} \]

Calculating the determinant:
\[ \vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} 2 & 3 \\ -2 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 3 & -2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \(\begin{vmatrix} 2 & 3 \\ -2 & 1 \end{vmatrix} = (2)(1) - (3)(-2) = 2 + 6 = 8\)
2. \(\begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} = (1)(1) - (3)(3) = 1 - 9 = -8\)
3. \(\begin{vmatrix} 1 & 2 \\ 3 & -2 \end{vmatrix} = (1)(-2) - (2)(3) = -2 - 6 = -8\)

Putting it all together:
\[ \vec{a} \times \vec{b} = 8\hat{i} + 8\hat{j} - 8\hat{k} = 8\hat{i} + 8\hat{j} - 8\hat{k} \]

### Step 4: Find the modulus of \(\vec{a} \times \vec{b}\)
\[ |\vec{a} \times \vec{b}| = \sqrt{(8)^2 + (8)^2 + (-8)^2} = \sqrt{64 + 64 + 64} = \sqrt{192} = 8\sqrt{3} \]

### Step 5: Calculate \(\sin \theta\)
Using the formula:
\[ \sin \theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|} \]
Substituting the values:
\[ \sin \theta = \frac{8\sqrt{3}}{\sqrt{14} \cdot \sqrt{14}} = \frac{8\sqrt{3}}{14} = \frac{4\sqrt{3}}{7} \]

### Final Answer
\[ \sin \theta = \frac{4\sqrt{3}}{7} \]