Finding the Cartesian Equation of a Perpendicular Line

[featheredteap]

Homework Statement

Find the vector equation of the line that passes through the point Q(2,0,-5) and
is perpendicular to both the vectors m=(0,1,4) and n=(-2,-1,3).

Homework Equations

vector equation of a line: (x, y, z)=(x₀,y₀,z₀) + t(a,b,c)
cartesian equation of a line: (x-x₀)/a=(y-y₀)/b=(z-z₀)/c

The Attempt at a Solution

(x₀,y₀,z₀)=(2,0,5)
To find (a,b,c)I know I can get two equations because the dot product of (a,b,c) with the two perpendicular lines equals zero:
b+4c=0
-2a-b+3c=0
But two equations isn't enough to solve for three variables. Also, shouldn't the point (2,0,-5) also dot product with u or v to equal zero, since it's on the same line?
Is it correct to assume that the points (0,1,4) and (-2,-1,3) are also points on the line? In which case I can easily find the direction of the line by subtracting one from the other.

 

[Mark44]

The cross product of <0, 1, 4> and <-2, -1, 3> will give you a vector that is perpendicular to both. That's the vector you need to write your line in either its vector form or in Cartesian form.

 

[featheredteap]

Man, I can't believe I didn't think of that myself. thanks for your help!