Question

Assertion(A): The sine of the angle between the vectors $\overrightarrow{a} = 4\hat{i} + 2\hat{j} - 3\hat{k}$ and $\overrightarrow{b} = 2\hat{i} - 5\hat{j} + 6\hat{k}$ is $\frac{\sqrt{2}}{7}$.

Reason(R): The angle $\theta$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is given by $\sin \theta = \frac{|\overrightarrow{a} \times \overrightarrow{b}|}{|\overrightarrow{a}||\overrightarrow{b}|}$.

• A. Both A and R are true and R is the correct explanation of A.
• B. Both A and R are true and R is not the correct explanation of A.
• C. A is true but R is false.
• D. A is false but R is true.

Answer 1

Answer: (D)

d

Answer 2

Reason (R) states that the angle $\theta$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is given by $\sin \theta = \frac{|\overrightarrow{a} \times \overrightarrow{b}|}{|\overrightarrow{a}||\overrightarrow{b}|}$. This is a standard formula derived from the definition of the magnitude of the cross product. Therefore, Reason (R) is true.

Assertion (A) states that the sine of the angle between the vectors $\overrightarrow{a} = 4\hat{i} + 2\hat{j} - 3\hat{k}$ and $\overrightarrow{b} = 2\hat{i} - 5\hat{j} + 6\hat{k}$ is $\frac{\sqrt{2}}{7}$. Calculating the cross product and magnitudes shows that this assertion is false.

Therefore, A is false but R is true.