Vectors

How to show points ae coplanar?

•  Find plane equation using 3 of the points

• Sub fourth point into equation and check if works

• r.n = a.n r.n̂ = p r · General point n · normal a · any point in plane

To find an equation of a plane with three points (a,b,c).

• Find ab, ac .

• dot each with (n̂), equals to 0

• let z=1, sim equations to get normal

• then use one point dotted with normal to get p

Intersection of line and plane

• Sub line into r.n = p

• find λ value

• sub λ into point

Parametric → Cartesian

• take n.b=0 n.c=0 from parametric (r=a+λb+μc)

• solve sim equations (z=1 method)

• Dot any point with n for p . r.n=p

Intersection of two planes (2 methods)

Method 1

• Find two common points on both planes (solve planes silmatenously - let z=0)

• find the vector through them, form line equation

Shortest distance between two skew lines

• Find gp's on both lines

• find vector between them (let t=λ-h) ensure the vector is perpendicular to l₁ ie. dot with one lines direction vector

• use value of t from that and put back into AB (vector between two)

• pythagoras for distance

Shortest distance between point and line

• Find gp on l₁

• vector between point and l₁

• ensure perp to l₁ (dot product=0)

• find λ sub into distance, pythagoras

Reflecting a point in a plane

• Intersection of line and plane (i.e Sub line into r.n=p , find λ value)

• Double λ value , sub into line

Reflecting a line in a plane

• Find intersection point (of plane and og line) -ie sub line in plane

• Reflect a point in the plane (direction of line is the normal btw) on l₁ to l₂

• Find equation of line with the 2 points