What is the sine of angle between the vectors `hat(i)+2hat(j)+3hat(k) and -hat(i)+2hat(j)+3 hat(k)`?

Options

A. `sqrt((13)/(7))`
B. `(sqrt(13))/(7)`
C. `(13)/(sqrt(7))`
D. None of these

Explanation

To find the sine of the angle between the vectors \( \hat{i} + 2\hat{j} + 3\hat{k} \) and \( -\hat{i} + 2\hat{j} + 3\hat{k} \), we can follow these steps:

### Step 1: Define the Vectors
Let:
\[ \mathbf{A} = \hat{i} + 2\hat{j} + 3\hat{k} \]
\[ \mathbf{B} = -\hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Calculate the Dot Product
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (1)(-1) + (2)(2) + (3)(3) \]
\[ = -1 + 4 + 9 = 12 \]

### Step 3: Calculate the Magnitudes of the Vectors
The magnitude of vector \( \mathbf{A} \) is:
\[ |\mathbf{A}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

The magnitude of vector \( \mathbf{B} \) is:
\[ |\mathbf{B}| = \sqrt{(-1)^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 4: Use the Cosine Formula
Using the formula for the cosine of the angle \( \theta \) between two vectors:
\[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]
Substituting the values we found:
\[ \cos \theta = \frac{12}{\sqrt{14} \cdot \sqrt{14}} = \frac{12}{14} = \frac{6}{7} \]

### Step 5: Find the Sine of the Angle
To find \( \sin \theta \), we can use the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can find \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{6}{7}\right)^2 = 1 - \frac{36}{49} = \frac{49 - 36}{49} = \frac{13}{49} \]
Now, taking the square root gives us:
\[ \sin \theta = \sqrt{\frac{13}{49}} = \frac{\sqrt{13}}{7} \]

### Final Answer
The sine of the angle between the vectors is:
\[ \sin \theta = \frac{\sqrt{13}}{7} \]