Determine the vector and parametric equations

[Styx]

Determine the vector and parametric equations of the plane that contains point C(1,-2,6) and the z-axis

I take this to mean that any point on the z-axis is valid so does that mean either (0, 0, 1) or (1, -2, 5) are also on the plane?

 

[pizzasky]

You are right in saying that the points (0, 0, 1) and (1, -2, 5) are on the plane.

However, can you see that the point (0, 0, 0) is also on the plane? (You don't really need to know this to solve your question, but I thought it would be nice if you are aware of this fact)

 

[Architeuthis Dux]

Yes, both (0, 0, 1) and (1, -2, 5) are also on the plane since they both lie on the z-axis.

To determine the vector and parametric equations of the plane containing point C(1,-2,6) and the z-axis, we first need to find the normal vector of the plane. The normal vector is a vector perpendicular to the plane, and it can be found by taking the cross product of two vectors that lie on the plane.

In this case, we can take the cross product of the vector (1, 0, 0) (a vector parallel to the x-axis) and the vector (0, 0, 1) (a vector parallel to the z-axis). The resulting cross product is the normal vector of the plane: (0, -1, 0).

Now, we can use the point-normal form of the equation of a plane to write the equation of the plane. The point-normal form is given by: N · (P - P0) = 0, where N is the normal vector, P is any point on the plane, and P0 is a known point on the plane.

Substituting in the values, we have: (0, -1, 0) · (P - (1, -2, 6)) = 0

Expanding this, we get: -y + 2 = 0

Solving for y, we get the equation of the plane: y = 2

This is the cartesian equation of the plane. To write the vector and parametric equations, we can use the following conversion:

Cartesian equation: ax + by + cz = d
Vector equation: r = P0 + t1v1 + t2v2
Parametric equations: x = x0 + t1a, y = y0 + t1b, z = z0 + t1c

In this case, our cartesian equation is y = 2. So, our vector equation becomes: r = (0, 2, 0) + t1(1, 0, 0) + t2(0, 0, 1)

And our parametric equations are: x = 0 + t1(1), y = 2 + t1(0), z = 0 + t1(0)

Simplifying,