Projection of a 3D Vector and Parallel Vectors

Qn1. If I have a 3D vector V(x,y,z). What will be the projection of vector V in

1.XY plane
2.YZ plane
3.ZX plane

Thanks in advance for answer.

Qn2. I have 3D points P1(x1,y1,z1), P2(x2,y2,z2), P3(x3,y3,z3), P4(x4,y4,z4). If–

x2-x1 = x4-x3
y2-y1 = y4-y3
z2-z1 = z4-z3

Does it mean that vector P1P2 and Vector P3P4 are parallel to each other.?





Projection of a vector to lets say XY plane would mean that you destroy its z component. Draw an arbitrary vector on space. Draw its x,y,z components (they should be nonzero for this example). Then For example take only x, y and add them. The resulting vector with no z component will be on what is so called xy plane which is the set of all points that can be defined using only x,y coordinates with z=0.

As for the second part, the difference you are taking there is actually finding the terms that define the slope of the two lines you defined. Taking two arbitrary points on a line it is slope on say xy direction is defined as (y2-y1)/(x2-x1). you can generalize that to zx and zy too. If its slope in all directions is the same then the two lines are either parallel OR they may be the same line. In addition to the conditions below, also if it is true that

“x3-x1=x2-x1 and y3-y1=y2-y1 and z3-z1=z2-z1”

then the two lines also (in addition to being parallel) lay on the same “axis”. In fact since lines are defined to be infinite to each side, they are the same lines.