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: Use a general point approach:
A point $\vec r$ lies in the plane if $\vec r-\vec a$ is a combination of the two given vectors, so $(\vec r-\vec a)\cdot(\vec u\times\vec v)=0$.

Step 2: Compute u x v:
$\vec u\times\vec v=-4\hat i-\hat j+5\hat k$ (from the determinant).

Step 3: Check which option matches:
The constant on the right side is $\vec a\cdot(\vec u\times\vec v)=26$, so $\vec r\cdot(-4\hat i-\hat j+5\hat k)=26$. Option D. As a check, $\vec u\cdot\vec n=-16+1+15=0$ and $\vec v\cdot\vec n=-4-1+5=0$.

Final Answer:
The normal is u x v = (-4, -1, 5) and r . n = 26. \[ \boxed{\text{(D) }\vec r\cdot(-4\hat i-\hat j+5\hat k)=26} \]