Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors and if $\frac{\pi}{4}$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$, then $$(\overrightarrow{a} + (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{b}) \cdot (\overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{b})$$ is:

• A. $\frac{1}{4}$
• B. $\frac{3}{4}$
• C. $\frac{3}{2}$
• D. $\frac{1}{2}$
• E. $\frac{5}{4}$

Hint:

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.

Answer 1

The Correct Option is D

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).