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Vector Operations

v[[i]] or Part[v,i]give the ith element in the vector v
c vscalar multiplication of c times the vector v
u.vdot product of two vectors
Norm[v]give the norm of v
Normalize[v]give a unit vector in the direction of v
Standardize[v]shift v to have zero mean and unit sample variance
Standardize[v,f1]shift v by f1[v] and scale to have unit sample variance

Basic vector operations.

This is a vector in three dimensions.
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This gives a vector u in the direction opposite to v with twice the magnitude.
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This reassigns the first component of u to be its negative.
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This gives the dot product of u and v.
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This is the norm of v.
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This is the unit vector in the same direction as v.
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This verifies that the norm is 1.
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Transform v to have zero mean and unit sample variance.
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This shows the transformed values have mean 0 and variance 1.
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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,v2,...}]generate an orthonormal set from the given list of vectors

Orthogonal vector operations.

This gives the projection of u onto v.
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p is a scalar multiple of v.
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u-p is orthogonal to v.
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Starting from the set of vectors {u, v}, this finds an orthonormal set of two vectors.
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When one of the vectors is linearly dependent on the vectors preceding it, the corresponding position in the result will be a zero vector.
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