Vector Operations

• The squared magnitude equals the dot-product of the vector with itself $|v{|}^2=v\cdot v$
• The dot product of two nonzero vectors satisfies $u\cdot v=|u||v|cos\theta$
• Vectors are orthogonal if $u\cdot v=0$. They are orthonormal if additionally $|u|=|v|=1$

Vectors in Three or More Dimensions

Underlies the Fourier Series and Wavelets

Properties

• $|u\pm v{|}^2=|u{|}^2\pm 2(u\cdot v)+|v{|}^2$
• If $u$ and $v$ are orthogonal, then $|u+v{|}^2=|u{|}^2+|v{|}^2$

Related

• Orthonormal Bases of Vectors