Multi - Lines, planes, points, distance, and intersections

Whether lines are identical, parallel, skew, or intersecting

1. Parallel: direction vectors have scalar multiple

2. Identical: plug in point from one line into equation for other

3. Set up system of equations for x, y, and z with parameters t and s, then substitute t and s into the third equation —> if true, intersecting, if false, skew

Distance between two lines

p1 and p2 are points on the line, v1 and v2 are the direction vectors of the lines

Distance between point and line

PQ is the vector formed from the given point and a point on the line

Distance between point and plane

point: (x1, y1, z1) plane: A(x-x0)+B(x-x0)+C(x-x0) = 0

Where line intersects plane

1. Find parametric equations of line

2. plug into plane equations

3. solve for t

4. plug t back into line equation

Line formed by the intersection of two planes

1. cross the normal vectors to find line vector

2. set one variable equal to zero in both plane equations, solving for the other two variables

Find plane equation from three points

1. Find vectors AB and BC, then cross them

2. plug resulting vector and any of the original points into the plane equation

Plane going through intersection of two planes and perpendicular to a third

1. write equation for family of planes

2. group the coefficients (rearrange)

3. if perpendicular = n1 dot n2 = 0

4. solve for lambda and substitute back

Dot product

perpendicular: cross product equals zero

Cross product

parallel: cross product equals zero

Scalar component

Vector projection