# Vector equation of line perpendicular to plane

• May 3rd 2010, 02:31 AM
axdv
Vector equation of line perpendicular to plane
Consider three points A,B,C with coordinates (2,2,3) (1,1,0) and (1,0,1) respectively. Find a vector equation of the line through point A that is perpendicular to the plane that passes through all three points A,B,C. [Hint: Use point-normal equation]

I am completely lost; the point normal equation bit throws me off the most.

Point normal equation: A(x-x0) + B(y-y0) + C(z-z0) = 0
• May 3rd 2010, 02:47 AM
HallsofIvy
Quote:

Originally Posted by axdv
Consider three points A,B,C with coordinates (2,2,3) (1,1,0) and (1,0,1) respectively. Find a vector equation of the line through point A that is perpendicular to the plane that passes through all three points A,B,C. [Hint: Use point-normal equation]

I am completely lost; the point normal equation bit throws me off the most.

Point normal equation: A(x-x0) + B(y-y0) + C(z-z0) = 0

I don't understand that hint. the "point normal equation" is the equation of a plane and you want to find a line, not a plane.

Instead, use the fact that the vector equation of a line through $\displaystyle (x_0, y_0, z_0)$ and in the direction of vector $\displaystyle A\vec{i}+ B\vec{j}+ C\vec{k}$ is $\displaystyle \vec{r}(t)= (At+ x_0)\vec{i}+ (Bt+ y_0)\vec{j}+ (Ct+ z_0)\vec{k}$.
You already know that $\displaystyle x_0= 2$, $\displaystyle y_0= 2$, $\displaystyle z_0= 3$ so you just need to find a vector perpendicular to the plane. And you can do that by taking the cross product of the vectors $\displaystyle (2- 1)\vec{i}+ (2- 1)\vec{j}+ (3- 0)\vec{k}$ (from B to A) and $\displaystyle (1- 1)\vec{i}+ (0- 1)\vec{j}+ (1- 0)\vec{k}$ (from B to C).

By the way, this is a "Calculus" problem. It doesn't belong in "Linear and Abstract Algebra".
• May 3rd 2010, 02:16 PM
axdv
Thanks for the help :)
And sorry about posting in the wrong place, I have difficulty telling different topics of maths appart.
• May 3rd 2010, 05:48 PM
Surreptitous_Smiles
Im confused
Can someone please explain that to me. Im a little lost.
• May 4th 2010, 03:20 AM
HallsofIvy
Quote:

Originally Posted by Surreptitous_Smiles
Can someone please explain that to me. Im a little lost.

Explain what to you?