Question

If \(\theta\) is the angle between the line \(\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})\) and the plane \(\bar{r}\cdot (\hat{i}-2\hat{j}+3\hat{k})=5\) then the value of \(\cos ^{2}\theta\) is

• A. \(\frac{6}{21}\)
• B. \(\frac{36}{42}\)
• C. \(\frac{8}{21}\)
• D. \(\frac{3}{21}\)

Hint:

The sine of the angle between a line and a plane uses the dot product of the direction with the plane normal; then use cos squared = 1 - sin squared.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
If \(\theta\) is the angle between a line with direction \(\bar b\) and a plane with normal \(\bar n\), then \(\sin\theta=\dfrac{|\bar b\cdot\bar n|}{|\bar b||\bar n|}\).

Step 2: Identify the Vectors
\(\bar b=\hat i-\hat j+\hat k\) and \(\bar n=\hat i-2\hat j+3\hat k\).

Step 3: Compute sin
\(\bar b\cdot\bar n=1+2+3=6\), \(|\bar b|=\sqrt{3}\), \(|\bar n|=\sqrt{14}\).
\(\sin^{2}\theta=\dfrac{36}{3\cdot 14}=\dfrac{36}{42}\).

Step 4: Find cos squared
\(\cos^{2}\theta=1-\dfrac{36}{42}=\dfrac{6}{42}=\dfrac{3}{21}\).

Final Answer:
The value of \(\cos^{2}\theta\) is \(\dfrac{3}{21}\). \[ \boxed{\text{(D) }\dfrac{3}{21}} \]