Distance of point to line equation
(Point: c, Line: a + λb)
Distance between parallel lines equation
(Points on lines: a1/a2, Direction Vector: b)
Distance between skew lines equation
(Points on lines: a1/a2, Direction Vectors: b1/b2)
How to work out the angle between two planes
Use scalar product to find the angle between the normals (therefore a= n1, b=n2)
How to work out the angle between a line and a plane
Use the scalar product to find the angle between the line's direction and the plane's normal, and subtract this value from 90°
How to work out line of intersection of 2 planes
The direction vector of the line of intersection is perpendicular to both plane's normals, therefore:
n1 x n2 = b
Then use simputaneous equations to find a point on both planes and this will be a, for line equation in form:
r = a + λb
Distance of point to plane
(Plane equation in form: r.n=d where n is the normal, r is a general point. Position of point: a)
Use equation: Distance=
Area of triange
Area of Parallelogram
Volume of Parallelepiped
Where a and b represent vector sides of the parallelogram at the base and c represets the vector upwards in the 3rd dimension:
Note: a.(bxc) = b.(cxa)
Volume of Tetrahedron
Same term definitions as volume of parallelepiped
Vector cross product formula
Where n stands for a unit vector perpendicular to both a and b. (in the direction that a right handed screw would move when turned a -> b)
Note: b x a = -a x b, and to find a unit vector, divide the vector by it's magnitude.
What is an alternate equation for a straight line in vector form?
(r-a) x b= 0
Can also be written as rxa - axb = 0 and hence as
What are the different forms for the eqaution of a plane?
(which can be converted to catesian form by inputting r= xi+yj+zk)
r= a + λb + μc
Where a is the position vector of a point in the plane, b and c are non-parallel vectors in the plane and r in the position vector a general point on the plane.
How to find the point of intersection of a line and a plane.
Enter the line in column matrix form and substitute for r in the plane's eqation, r.n=a.n .