Question

The vector equation of plane in parametric form, passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line \(\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}\) is

• A. \(\overset{̄}{r} = (-\hat{i}+2\hat{j})+λ(3\hat{i}-\hat{k})+μ(\hat{i}+\hat{j}-\hat{k})\)
• B. \(\overset{̄}{r} = (-\hat{i}+2\hat{j})+λ(3\hat{i}-\hat{k})+μ(\hat{i}+2\hat{j}-\hat{k})\)
• C. \(\overset{̄}{r} = (\hat{i}-2\hat{j})+λ(3\hat{i}+\hat{k})+μ(\hat{i}+\hat{j}-\hat{k})\)
• D. \(\overset{̄}{r} = (-\hat{i}+2\hat{j})+λ(3\hat{i}-\hat{k})+μ(\hat{i}-\hat{j}+\hat{k})\)

Hint:

Use one point as position vector, the line joining the two points as one direction, and the given line as the other.

Answer 1

The Correct Option is A

Step 1: Direction vectors
The line joining \((-1,2,0)\) and \((2,2,-1)\) has direction \(3\hat i - \hat k\).

Step 2: Given line
\(\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}\) means \(\frac{x-1}{1} = \frac{y+\frac12}{1} = \frac{z+1}{-1}\), so its direction is \(\hat i+\hat j-\hat k\).

Step 3: Write the plane
With position vector \(-\hat i+2\hat j\): \(\vec r = (-\hat i+2\hat j)+\lambda(3\hat i-\hat k)+\mu(\hat i+\hat j-\hat k)\). Option (A).

Step 4: Common mistake
Option (B) uses \(\hat i+2\hat j-\hat k\), which forgets to divide \(2y\) by 2.

Final Answer:
The plane is r = (-i + 2j) + lambda(3i - k) + mu(i + j - k). \[ \boxed{\text{(A)}\ \vec r=(-\hat i+2\hat j)+\lambda(3\hat i-\hat k)+\mu(\hat i+\hat j-\hat k)} \]