Vector parallel to the line
 

Hi Guys! I was doing a vector math problem and I got stuck on how to find the vector parallel to the line.
The question is : Determine if the plane given by -x+2z=10 and the line given by r=(5,2-t,10+4t) are orthogonal, parallel or neither.

Now I have already found the normal vector as the first step, but also I have to find the vector that is parallel to the line so as that I can cross it with the normal vector to prove if it they are parallel to each other.
Thank you.

Is there a point that lies in the given plane and on the given line?

I presume that you have determined that a normal vector to the plane is <-1, 0, 2>. Is that what you got? Since your line is given by r = (5, 2 - t, 10 + 4t), taking t = 0, one point on that line is (5, 2, 10). Taking t = 1, another point is (5, 1, 14). The vector from (5, 2, 10) to (5, 1, 14) is <5 - 5, 1 - 2, 14 - 10> = <0, -1, 4> and that is a vector parallel to the line. Notice that <0, -1, 4> are the coefficients of t in "r = 5, 2 - t, 10 + 4t".

In general, a vector in the direction of the line $\displaystyle x= x_0+ At$, $\displaystyle y= y_0+ Bt$, $\displaystyle z= z_0+ Ct$ is <A, B, C>.

Thank you very much Country Boy, you saved me in this. I now get the full picture. Thank you again.
#Slumerican

A tad faster might be:
A normal to plane is (-1,0,2) by inspection.
Line is r=(5,2,10)+t(0,-1,4)
(-1,0,2)x(0,-1,4)=0

[Note by moderator: (-1, 0, 2) × (0, -1, 4) = (2, 4, 1)]