Question

If \(\theta\) is the angle between the vectors \(2 \hat{i} - 2 \hat{j} +4 \hat{k}\) and \(3 \hat{i} + \hat{j} + 2 \hat{k}\), then value of sin \(\theta\) is:

• A. \(\frac{2}{3}\)
• B. \(\frac{2}{\sqrt {7}}\)
• C. \(\frac{\sqrt {2}}{7}\)
• D. \(\sqrt{\frac{2}{7}}\)

Answer 1

The Correct Option is B

To find the value of \( \sin \theta \) for the angle \( \theta \) between the vectors \(\mathbf{A} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k}\) and \(\mathbf{B} = 3 \hat{i} + \hat{j} + 2 \hat{k}\), we use the formula: \(\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}\).

Step 1: Calculate the dot product \(\mathbf{A} \cdot \mathbf{B}\).

\[ \mathbf{A} \cdot \mathbf{B} = (2)(3) + (-2)(1) + (4)(2) \]

\[ = 6 - 2 + 8 \]

\[ = 12 \]

Step 2: Calculate the magnitudes of \(\mathbf{A}\) and \(\mathbf{B}\).

\[ |\mathbf{A}| = \sqrt{(2)^2 + (-2)^2 + (4)^2} \]

\[ = \sqrt{4 + 4 + 16} \]

\[ = \sqrt{24} = 2\sqrt{6} \]

\[ |\mathbf{B}| = \sqrt{(3)^2 + (1)^2 + (2)^2} \]

\[ = \sqrt{9 + 1 + 4} \]

\[ = \sqrt{14} \]

Step 3: Substitute values into the cosine formula and solve for \(\sin \theta\).

\[ \cos \theta = \frac{12}{2\sqrt{6} \cdot \sqrt{14}} \]

\[ = \frac{12}{2\sqrt{84}} \]

\[ = \frac{6}{\sqrt{84}} \]

To find \(\sin \theta\), use the identity:

\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \]

\[ \sin \theta = \sqrt{1 - \left(\frac{6}{\sqrt{84}}\right)^2} \]

\[ = \sqrt{1 - \frac{36}{84}} \]

\[ = \sqrt{\frac{48}{84}} \]

\[ = \sqrt{\frac{4}{7}} \]

\[ = \frac{2}{\sqrt{7}} \]

Therefore, the value of \(\sin \theta\) is \(\frac{2}{\sqrt{7}}\).