Where does the line [x, y, z] = [6, 10, 1] + t[3, 4, -1] meet the xy-plane?

The x-y plane is where z=0.

So solve the equation of the line where z=0, i.e.
z=1+t[-1]=0
=>
t=1
Substitute t=1 into
[x, y, z] = [6, 10, 1] + t[3, 4, -1]
gives
[x,y,z]=[6,10,1]+1[3,4,-1]=[9,14,0]

So the final answer for this one would be

[x,y,z]=[6,10,1]+1[3,4,-1]=[9,14,0] ?

Thank you!

Right.

You're welcome!

To find where the line [x, y, z] = [6, 10, 1] + t[3, 4, -1] meets the xy-plane, we need to solve for t when z = 0.

Step 1: Set z = 0
0 = 1 + t*(-1)
0 = 1 - t

Step 2: Solve for t
t = 1

Step 3: Substitute t back into the equation
[x, y, z] = [6, 10, 1] + 1[3, 4, -1]
[x, y, z] = [6, 10, 1] + [3, 4, -1]
[x, y, z] = [6 + 3, 10 + 4, 1 - 1]
[x, y, z] = [9, 14, 0]

Therefore, the line [x, y, z] = [6, 10, 1] + t[3, 4, -1] meets the xy-plane at the point (9, 14, 0).