Vector lines and planes

Vector Line equations given a point and direction

l=a+td

Vector line equation given 2 points

l=a+t(b-a)

Vector line to Cartesian equation (2 dimensions)

1. Split vector into i and j components

2. Equate each component to x and y. There will be 3 variables - x, y, and t

3. Solve the x equation for t, so you have t in terms of x

4. Sub this equation of t into y

Vector line to Cartesian equation (3 dimensions)

1. Split vector into i and j components

2. Equate each component to x, y, and z

3. Solve each one for t, so you have t in terms of x, y, and z

4. Equate them all, so you have …x=…y=…z

Cartesian equation to vector line (2 dimensions)

1. Find any point on the line - try x or y = 0

2. Convert the point to a vector. This will become a.

3. Take the gradient - m - and turn it into a vector using rise/run, run =1 i.e. gradient of 2 turns into i + 2j. The line is parallel to this vector, so this becomes d. DONE!

Cartesian equation into vector line (3 dimensions)

1. Let each part of the equation equal t

2. Solve for x, y, z in terms of t

3. Equate them for i, j, and k

Shortest distance from a point to a line

1. Let point A equal the point, point P equal a point on the line

2. Distance from A to P is AP, find AP=P-A

3. Solve AP·d =0 to find t where they are perpendicular

4. Sub t into AP

5. Find |AP|

Point on a line such that the distance to a point is the shortest

1. Let point A equal the point, point P equal a point on the line

2. Distance from A to P is AP, find AP=P-A

3. Solve AP·d =0 to find t where they are perpendicular

4. Sub t into P

Intersection of 2 lines

equate i,j,k to find the parameter(s) - check with all equations. Sub back into equation to find value