Question

The vector equation of the plane passing through the point \(A(-2,7,5)\) and parallel to the vectors \(4\hat{i}-\hat{j}+3\hat{k}\) and \(\hat{i}+\hat{j}+\hat{k}\) is ...

• A. \(\overset{̄}{r}\cdot (-2\hat{i}+7\hat{j}+5\hat{k}) = 26\)
• B. \(\overset{̄}{r}\cdot (4\hat{i}-\hat{j}+3\hat{k}) = 27\)
• C. \(\overset{̄}{r}\cdot (4\hat{i}-\hat{j}+5\hat{k}) = 26\)
• D. \(\overset{̄}{r}\cdot (-4\hat{i}-\hat{j}+5\hat{k}) = 26\)

Hint:

The normal is the cross product of the two vectors; then use r . n = a . n.

Answer 1

The Correct Option is D

Step 1: Find the normal:
\(\vec n=(4\hat i-\hat j+3\hat k)\times(\hat i+\hat j+\hat k)=\begin{vmatrix}\hat i&\hat j&\hat k\\4&-1&3\\1&1&1\end{vmatrix}\)
\(=\hat i(-1-3)-\hat j(4-3)+\hat k(4+1)=-4\hat i-\hat j+5\hat k\).

Step 2: Compute a . n:
Position vector of A is \(\vec a=-2\hat i+7\hat j+5\hat k\). \(\vec a\cdot\vec n=(-2)(-4)+7(-1)+5(5)=8-7+25=26\).

Step 3: Write the equation:
\(\vec r\cdot(-4\hat i-\hat j+5\hat k)=26\). Option D.

Step 4: Why the other options are wrong.
Option A uses the position vector of A as the normal. Option B uses one of the parallel vectors as the normal. Option C has the right form but the wrong x-component of the normal.

Final Answer:
The plane is r . (-4i - j + 5k) = 26. \[ \boxed{\text{(D) }\vec r\cdot(-4\hat i-\hat j+5\hat k)=26} \]