Question

The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x + y - 2z = 18$ are

• A. $(4, 2, -4)$
• B. $(1, 2, -3)$
• C. $(4, 2, 4)$
• D. $(4, -2, -4)$

Hint:

The foot of the perpendicular must reside directly inside the target plane. You can test the options by plugging them directly into the plane equation $2x + y - 2z = 18$. For option (A): $2(4) + 2 - 2(-4) = 8 + 2 + 8 = 18$. Since it satisfies the equation and its components are directly proportional to the normal vector direction ratios $(2,1,-2)$, it must be the correct point!

Answer 1

The Correct Option is A

Step 1: Picture the setup.
We drop a perpendicular from the origin to the plane $2x + y - 2z = 18$; the foot lies along the plane's normal direction.
Step 2: Read the normal vector.
The coefficients give the normal $(2, 1, -2)$.
Step 3: Parametrise the perpendicular line.
A point on the line from the origin is $(2t,\ t,\ -2t)$ for some scalar $t$.
Step 4: Force the point onto the plane.
Substitute into the plane: $2(2t) + (t) - 2(-2t) = 18$, i.e. $4t + t + 4t = 18$, so $9t = 18$.
Step 5: Solve for t.
$t = 2$.
Step 6: Find the foot.
The coordinates are $(2t, t, -2t) = (4, 2, -4)$, which is option 1 and matches the key.
\[ \boxed{(4,\ 2,\ -4)} \]