Mathematica Tutorial

Vector Operations

v[[i]] or Part[v,i]	give the i^th element in the vector v
c v	scalar multiplication of c times the vector v
u.v	dot product of two vectors
Norm[v]	give the norm of v
Normalize[v]	give a unit vector in the direction of v

Basic vector operations.

This is a vector in three dimensions.

This gives a vector u in the direction opposite to v with twice the magnitude.

This reassigns the first component of u to be its negative.

This gives the dot product of u and v.

This is the norm of v.

This is the unit vector in the same direction as v.

This verifies that the norm is 1.

Two vectors are orthogonal if their dot product is zero. A set of vectors is orthonormal if they are all unit vectors and are pairwise orthogonal.

Projection[u,v]	give the orthogonal projection of u onto v
Orthogonalize[{v₁,v_2,...}]	generate an orthonormal set from the given list of vectors

Orthogonal vector operations.

This gives the projection of u onto v.

p is a scalar multiple of v.

u-p is orthogonal to v.

Starting from the set of vectors {u, v}, this finds an orthonormal set of two vectors.

When one of the vectors is linearly dependent on the vectors preceding it, the corresponding position in the result will be a zero vector.