Surfaces
z=f(x,y)Describing surfaces geometrically:ex: "x=-1 is parallel to the yz-plane, back 1 unit"ex: "x+y=2 is a vertical plane intersecting with x+y=2 on the xy-plane
Distance Formula (3D Points)
Same process as finding the vector between the two points, and then finding the vector's magnitude.
Equation of a Sphere
Needs a point (h, k, l) and the sphere's radius r. When they only give you two points, find the distance between them for the radius.
Vectors
Only have (1) direction and (2) magnitude/length. Head is the arrowhead side.The components of vector <x,y, z> are x, y, and z.
Vector Addition
0
Scalar Multiplication
Multiply each component of a vector by the scalar/number. A negative scalar makes the vector go in the opposite direction.
Head-Tail Rule
To make a vector between two points. Subtract the tail point from the head point.
Normalizing the Vector
Making the vector have a magnitude of 1. Divide the vector by its magnitude, or with scalar multiplication, multiply the vector with 1/magnitude.
Dot Product
Gives a scalar result.
Finding the Angle between Two Vectors
Dot product in the numerator.If the numerator is 0, then the angle is 90 degrees and the vectors are perpendicular/orthogonal.If the numerator is 1, then the angle is 0 and the vectors are parallel.If the numerator is -1, then the angle is pi and the vectors are anti-parallel.
Vector Projection
The projection of vector u onto v, and this projection is called w1.w1 and w2 are vector components of u, so u=w1+w2, and are also perpendicular/orthogonal to each other.
Scalar Projection
The magnitude of the vector projection/w1.
Cross Product
The cross product of two vectors produces a vector perpendicular to the original two vectors.
Magnitude of Cross Product
Equal to the area of the parallelogram formed by the two vectors.
Vector Form of a Line
Needs (1) a point on the line and (2) a vector parallel to the line.The (1) point is made into a vector and added to the (2) parallel vector which is multiplied by scalar 't', and the sum produces a vector that describes a point on the line.
Scalar Equation of a Plane
Needs (1) a vector normal/perpendicular to the plane and (2) a point on the plane.