Question

Consider the line which passes through the point P(-1, 1, -4), and which is parallel to the line x=1+7t, y=2+3t, z=3+3t How do you find the point of intersection of this new line with each of the coordinate planes?

Answer 1

The intersection of a line passing through the point P(-1, 1, -4) parallel to another line with the coordinate planes is found by setting each coordinate to zero and solving for the parametric variable t.

Step-by-step explanation:

Finding the point of intersection of a line with the coordinate planes involves using the parametric equations of the line.

In this case, the line passing through point P(-1, 1, -4) which is parallel to the line x=1+7t, y=2+3t, z=3+3t can be represented in parametric form as:

x = -1 + 7t
y = 1 + 3t
z = -4 + 3t

To find the intersection with the coordinate planes, we'll set each of the x, y, and z components to zero one at a time and solve for t to find the corresponding points.

For the xy-plane (z=0), set z = -4 + 3t = 0 and solve for t. This gives t = 4/3, and substituting back gives x = -1 + 7*(4/3) and y = 1 + 3*(4/3), resulting in the intersection point with the xy-plane.

Repeat this process for the other planes:

For the xz-plane (y=0), set y = 1 + 3t = 0.

For the yz-plane (x=0), set x = -1 + 7t = 0.