Question

Consider the following in respect of the vectors $\overrightarrow{a} = (1,0,0)$ and $\overrightarrow{b} = (1, 1, 1)$:

1. The projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is $\frac{1}{\sqrt{3}}$.

2. The angle between the vectors is $\frac{\pi}{3}$.

Which of the statements given above is/are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (A)

1 only

Answer 2

Statement 1: The projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is $\frac{1}{\sqrt{3}}$.

The projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$ is given by the formula: $Proj_{\overrightarrow{b}} \overrightarrow{a} = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{||\overrightarrow{b}||}$

First, calculate the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$: $\overrightarrow{a} \cdot \overrightarrow{b} = (1)(1) + (0)(1) + (0)(1) = 1 + 0 + 0 = 1$

Next, calculate the magnitude of vector $\overrightarrow{b}$, $||\overrightarrow{b}||$: $||\overrightarrow{b}|| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{1 + 1 + 1} = \sqrt{3}$

Now, substitute these values into the projection formula: $Proj_{\overrightarrow{b}} \overrightarrow{a} = \frac{1}{\sqrt{3}}$

Thus, Statement 1 is correct.

Statement 2: The angle between the vectors is $\frac{\pi}{3}$.

The cosine of the angle $\theta$ between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is given by the formula: $\cos \theta = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{||\overrightarrow{a}|| \cdot ||\overrightarrow{b}||}$

We already calculated $\overrightarrow{a} \cdot \overrightarrow{b} = 1$ and $||\overrightarrow{b}|| = \sqrt{3}$.

Next, calculate the magnitude of vector $\overrightarrow{a}$, $||\overrightarrow{a}||$: $||\overrightarrow{a}|| = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1$

Now, substitute these values into the formula for $\cos \theta$: $\cos \theta = \frac{1}{(1)(\sqrt{3})} = \frac{1}{\sqrt{3}}$

To check if the angle is $\frac{\pi}{3}$, we compare $\cos \theta$ with $\cos\left(\frac{\pi}{3}\right)$: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

Since $\frac{1}{\sqrt{3}} \neq \frac{1}{2}$ (as $\sqrt{3} \neq 2$), the angle between the vectors is not $\frac{\pi}{3}$. The actual angle is $\theta = \arccos\left(\frac{1}{\sqrt{3}}\right)$.

Thus, Statement 2 is incorrect.

Conclusion: Statement 1 is correct, and Statement 2 is incorrect. Therefore, only Statement 1 is correct.