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: Use the coplanarity condition:
The lines meet if they are coplanar and not parallel: the vector joining the points $(4 - 1, 1 - m, 0 - 3) = (3, 1 - m, -3)$ must satisfy a determinant condition with the directions $(2, 3, 4)$ and $(5, m, 1)$.

Step 2: Evaluate:
\[ \begin{vmatrix}3 & 1 - m & -3\\2 & 3 & 4\\5 & m & 1\end{vmatrix} = 3(3 - 4m) - (1 - m)(2 - 20) - 3(2m - 15) \]
\[ = 9 - 12m + 18(1 - m) - 6m + 45 = 72 - 36m = 0 \Rightarrow m = 2 \]
Option (C).

Final Answer:
$m = 2$. \[ \boxed{2} \]