Math vectors

Intersection between two planes (2 steps)

Finding a line.

Do scalar/vector product of normal of both planes to get the direction vector of the line

Pick/Find a point that is on both of the planes and use that as position vector

Intersection between two lines (3 Steps)

Make sure the lines use different parameters

Equate the generic points (x’s, y’s, and z’s) of both lines.

Solve using two of the equations and confirm with the third unused one

Intersection between line and plane (3 Steps)

Take generic point of line, meaning x,y,z including parameter

Plug the generic point x,y,z of line into equation of plane and solve for parameter

Plug parameter into equation of line to get the point that is on the line and plane.

How to determine if two lines are parallel (2 ways)

If dot product is 1, or if direction vectors are multiples or each other.

Distance between (two parallel lines). (6 Steps & Similar to line and point)

Pick a point on line A

Use given equation to name a general point on line B

Find vector directly perpendicular AND between both lines by subtracting the points.

Do cross product is = 0. Rearrange and solve for the parameter.

EITHER:

Plug parameter into line b to find where in line b that point lies.

Then use distance between two points formula.

OR

Plug parameter into vector directly perpendicular and between both lines and calculate magnitude

Distance between point and plane (4 Steps)

Use normal vector of plane to create direction vector

Use point to create position vector and complete the line equation

Plug generic point x,y,z of the line equation into the plane equation and solve for parameter

Plug parameter value into the derived line to get the vector then do magnitude of that.

Distance between two lines (5 Steps)

Do not use the same parameter for the two lines

Do cross/vector product of the two direction vectors to get direction vector perpendicular to both lines

Use generic point of both planes and use create a new parameter which you multiply with direction vector to reach from one plane to the other.

Do system of linear equations to solve for the new parameter

Plug that parameter into equation and find magnitude to get distance

Distance between a point and line. Similar to two parallel lines (5 Steps)

Use given equation to name a general point on line B

Find vector directly perpendicular to line and point by subtracting general point of line from the given point.

Do cross product is = 0. Rearrange and solve for the parameter.

Plug parameter into line to find where in line b that point lies.

Then use distance between two points formula.

Distance between two parallel planes (5 Steps)

Get normal of the plane and use as direction vector

Get point on: the other plane or the line parallel to that plane. Use as position vector

Plug the generic point x,y,z of line into equation of plane of unknown point and solve for parameter

Plug parameter into equation of line to get the point that is on the line and plane.

Use distance between two points formula.

Angle between two lines

Do dot product of direction vectors

Angle between two planes

Simply dot product of normal of planes

Angle a plane and a line

Dot product of normal of plane and line

Then do pi/2 - that angle to get the ACTUAL ANGLE

If a line and a plane have no points in common it means???

They must be parallel to each other and dot product of line and normal of plane is 0