If angle between ` veca = hati - 2hatj + 3hatk` and `vecb = 2hati + hatj +hatk` is `theta` then the value of `sin theta ` is

Options

A. `(3)/(2sqrt7)`
B. `(-2)/(sqrt7)`
C. `(4)/(3sqrt7)`
D. `(5)/(2sqrt7)`

Answer

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

### Step 1: Calculate the dot product \( \vec{a} \cdot \vec{b} \)

The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:

\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

For our vectors:

- \( \vec{a} = (1, -2, 3) \)
- \( \vec{b} = (2, 1, 1) \)

Calculating the dot product:

\[ \vec{a} \cdot \vec{b} = (1)(2) + (-2)(1) + (3)(1) = 2 - 2 + 3 = 3 \]

### Step 2: Calculate the magnitudes of \( \vec{a} \) and \( \vec{b} \)

The magnitude of a vector \( \vec{v} = (x, y, z) \) is given by:

\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]

Calculating the magnitude of \( \vec{a} \):

\[ |\vec{a}| = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

Calculating the magnitude of \( \vec{b} \):

\[ |\vec{b}| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

### Step 3: Use the dot product to find \( \cos \theta \)

The cosine of the angle \( \theta \) between the two vectors is given by:

\[ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \]

Substituting the values we found:

\[ \cos \theta = \frac{3}{\sqrt{14} \cdot \sqrt{6}} = \frac{3}{\sqrt{84}} = \frac{3}{2\sqrt{21}} \]

### Step 4: Use the Pythagorean identity to find \( \sin \theta \)

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ \sin^2 \theta = 1 - \cos^2 \theta \]

Calculating \( \cos^2 \theta \):

\[ \cos^2 \theta = \left(\frac{3}{2\sqrt{21}}\right)^2 = \frac{9}{4 \cdot 21} = \frac{9}{84} = \frac{3}{28} \]

Now substituting back into the identity:

\[ \sin^2 \theta = 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]

Taking the square root to find \( \sin \theta \):

\[ \sin \theta = \sqrt{\frac{25}{28}} = \frac{5}{\sqrt{28}} = \frac{5}{2\sqrt{7}} \]

### Final Answer

Thus, the value of \( \sin \theta \) is:

\[ \sin \theta = \frac{5}{2\sqrt{7}} \]