Product of Vector in Algebra (Dot Product)

If ā and b̄ are any two vectors, then the scalar product of these vectors is

ā · b̄ = |ā| |b̄| cos θ = ab cos θ, where θ is the angle between ā and b̄.

\[\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\]

Special Cases

• Perpendicular → \[\overline{\mathrm{a}}\cdot\overline{\mathrm{b}}=0\]

• Parallel → \[\mathbf{\overline{a}}\cdot\mathbf{\overline{b}}=\mathbf{ab}\]

Projection

• Scalar= \[\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}\]

• Vector \[=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^{2}}\cdot\mathbf{b}\]