Question

determine a vector equation for each line.
c) parallel to the z-axis and through P(1,5,10)
d)parallel to [x,y,z]=[3,3,0]+t[3,-5,-9] with the x-int -10
e)With the same x-intercept as [x,y,z]=[3,0,0]+t[4,-4,1] and the same z-intercept as [x,y,z]=[6,-2,3]+t[3,-1,-2]

Answer 1

Use the respective formulas for the Question .

Answer 2

For c) , it needs to be parallel to the z axis, so you may use the vector k. It also says it has to cross through (1,5,10), so the equation can be found easily:

\[p=(1,5,10)+t(0,0,1)\]

Answer 3

oh ok thanks. What about e)

Answer 4

well, for the first line, look at the point you're given: (3,0,0). That's exactly where the line crosses the x axis.

For the second line, you have to make a system of linear equations, making x and y zero, to find the z coordinate:

6+3t=0 => t=-2
-2-t=0 => t=-2
3-2t=z

Since the values for t are equal, it means the line crosses the z axis at t=-2, so now substitute that value in the third equation:

3-2(-2)=z ; z=7

So the second line intercepts the z axis at (0,0,7). Finally, to find the equation you're asked for, do a vector substraction:

(3,0,0)-(0,0,7)=(3,0,-7)

And use any of the points found earlier. So finally, the equation would look like this:

[x,y,z]=(3,0,0)+t(3,0,-7)