NIMCET 2009 — Mathematics PYQ

Using the standard expansion formula for the vector triple product:

2. Equate to the Given Expression

From the problem in image_73b17e.png, we are given:

3. Compare Coefficients

Since $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are non-coplanar, we can compare the coefficients of vectors $\mathbf{b}$ and $\mathbf{c}$ on both sides of the equation:

• Coefficient of $\mathbf{b}$: $\mathbf{a}\cdot \mathbf{c}=\frac{1}{\sqrt{2}}$

• Coefficient of $\mathbf{c}$: $-(\mathbf{a}\cdot \mathbf{b})=\frac{1}{\sqrt{2}}⟹\mathbf{a}\cdot \mathbf{b}=-\frac{1}{\sqrt{2}}$

4. Find the Angle between $\mathbf{a}$ and $\mathbf{b}$

Let the angle between $\mathbf{a}$ and $\mathbf{b}$ be $\theta$. Since they are unit vectors ($|\mathbf{a}|=1,|\mathbf{b}|=1$):

5. Determine the Value of $\theta$

For $\cos \theta =-\frac{1}{\sqrt{2}}$, the angle in the standard range $[0,\pi ]$ is:

Final Answer:

The correct option is (b) $\frac{3\pi }{4}$.