Question

For the line \(\frac{x+1}{1} = \frac{y-2}{2} = \frac{z+3}{3}\), identify the incorrect statement among the following.

• A. It can be represented by equation \(\frac{x+2}{1} = \frac{y}{2} = \frac{z+6}{3}\)
• B. It lies in the plane \(x-2y+z+8 = 0\)
• C. It is perpendicular to the plane \(x-2y+z = 0\)
• D. It passes through point \((0,4,0)\).

Hint:

Perpendicular to a plane means parallel to its normal.

Answer 1

The Correct Option is C

Step 1: Parameter form:
Points on the line are $(-1+t,\,2+2t,\,-3+3t)$.

Step 2: Test (D) and (A):
$t=1$ gives $(0,4,0)$. $t=-1$ gives $(-2,0,-6)$, which is the base point of the line in (A).

Step 3: Test (C):
Perpendicular to a plane means parallel to its normal. $(1,2,3)\times(1,-2,1)=(8,2,-4)\ne0$, so not parallel.

Step 4: Test (B):
Every point $(-1+t,2+2t,-3+3t)$ gives $x-2y+z=-1+t-4-4t-3+3t=-8$, so $x-2y+z+8=0$ always. True.

Final Answer:
The cross product of direction and normal is not zero. \[ \boxed{C} \]