What are the considerations when finding vector equations for lines and planes?

[suspenc3]

A few Questions:

a)when finding vector equations (for lines), what do you different when they give you a vector and a line parallel to this vector, and a vector and a line perpendicular to the vector.

b)concerning planes, can someone briefly explain the normal vector.

Thanks.

 

[radou]

A few Questions:

a)when finding vector equations (for lines), what do you different when they give you a vector and a line parallel to this vector, and a vector and a line perpendicular to the vector.

b)concerning planes, can someone briefly explain the normal vector.

Thanks.

I'll try to help with b).

The easiest way to copletely determine a plane is with one point T1 = (x1, y1, z1) (belonging to the plane) and a vector n which is perpendicular to the plane, which we call the normal vector. Now, let T = (x, y, z) be any point in the plane. Obviously, n must be perpendicular to the vector $$\vec{T_{1}T}$$, which implies n(r-r1)=0 ...(1), where r1 is the radius vector determined by the point T1, and r the radius vector determined by the point T. Further on, (1) directly implies A(x-x1) + B(y-y1)+ C(z-z1) = 0, where n=Ai+Bj+Ck. This is a general equation of a plane.

 

[suspenc3]

Yeah that explains it, so say they give you a point and a vector parallel to the plane, how would you get the Normal? Would you just cross the vector and the point?

 

[radou]

Yeah that explains it, so say they give you a point and a vector parallel to the plane, how would you get the Normal? Would you just cross the vector and the point?

A point and a vector parallel to a plane do not determine a plane. They determine an infinite number of planes.