Kerala KEAM 5 June 2024 Solved Paper

We are given two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, and the angle between them is $\frac{\pi}{4}$. The formula for the dot product of two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ is:$$\overrightarrow{u} \cdot \overrightarrow{v} = | \overrightarrow{u} | \overrightarrow{v} | \cos \theta.$$Since $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, their magnitudes are both 1, so: $$\overrightarrow{a} \cdot \overrightarrow{b} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}.$$Now, we need to calculate the expression: $$(\overrightarrow{a} + (\overrightarrow{a} \cdot \overrightarrow{b}) \overrightarrow{b}) \cdot (\overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{b}) \overrightarrow{b}).$$Substitute $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{\sqrt{2}}$ into the expression: $$(\overrightarrow{a} + \frac{1}{\sqrt{2}} \overrightarrow{b}) \cdot (\overrightarrow{a} - \frac{1}{\sqrt{2}} \overrightarrow{b}).$$Use the distributive property of the dot product: $$\overrightarrow{a} \cdot \overrightarrow{a} - \frac{1}{\sqrt{2}} \overrightarrow{a} \cdot \overrightarrow{b} + \frac{1}{\sqrt{2}} \overrightarrow{b} \cdot \overrightarrow{a} - \frac{1}{2} \overrightarrow{b} \cdot \overrightarrow{b}.$$Since $\overrightarrow{a} \cdot \overrightarrow{a} = 1$ and $\overrightarrow{b} \cdot \overrightarrow{b} = 1$, and $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{\sqrt{2}}$, the expression becomes: $$1 - \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} - \frac{1}{2}.$$Simplify: $$1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{2} = \frac{1}{2}.$$Thus, the value of the expression is $\frac{1}{2}$, and the correct answer is option (D).Quick Tip: When calculating the dot product of expressions involving unit vectors, use the fact that their magnitudes are 1 and apply the properties of dot products to simplify the calculations.