Consider the line which passes through the point P(-4,-2,-2), and which is parallel to the linex=1+7t,y=2+6t,z=3+5t Find the point of intersection of this new line with each of the coordinate planes:

Question

Find the point of intersection of this new line with each of the coordinate planes: xy-plane ( __, __, __) xz-plane (__, __, __) yz-plane (__,__,__)

Answer 1

The tangent vector along the given line is T =(dx/dt, dy/dt, dz/dt) = (7, 6, 5).

You need a line through P(-4,-2,-2) with that tangent vector (or parallel to it).

An easy parametrization of the line is:

(x, y, z) = P + T.t = (-4 + 7t, -2 + 6t, -2 + 5t)

All points in the xy plane have z = 0 so set z = -2 + 5t = 0, solve for t and find the other two coordinates x and y by plugging that value of t in their equations. The point of intersection will be (x, y, 0).

Analogously for the intersection points with the other two planes.