Question

The angle between the line \(x-1 = 2-y = \frac{2z-6}{4}\) and the plane \(\overset{⃗}{r}\cdot (2\hat{i}+\hat{j}+\hat{k}) = 10\) is

• A. \(\frac{π}{3}\)
• B. \(\frac{π}{6}\)
• C. \(\frac{π}{4}\)
• D. \(\frac{π}{12}\)

Hint:

Equate the coordinates of the two lines and solve for the parameters and m.

Answer 1

The Correct Option is B

Step 1: Equate components:
Line 1: \((1 + 2\lambda,\ m + 3\lambda,\ 3 + 4\lambda)\). Line 2: \((4 + 5\mu,\ 1 + m\mu,\ \mu)\).

Step 2: Solve for lambda and mu:
From the z components: \(\mu = 3 + 4\lambda\). From the x components: \(1 + 2\lambda = 4 + 5(3 + 4\lambda) = 19 + 20\lambda\), so \(\lambda = -1\) and \(\mu = -1\).

Step 3: Use the y components:
\(m + 3\lambda = 1 + m\mu \Rightarrow m - 3 = 1 - m \Rightarrow 2m = 4 \Rightarrow m = 2\).

Final Answer:
\(m = 2\), option (C). \[ \boxed{2} \]