Compute dot product, cross product, magnitude, unit vector, angle between vectors, projections, and triple products — with complete step-by-step working.
Dimension
2D (x, y)
3D (x, y, z)
Dot Product: $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta$
Cross Product (3D only): $\mathbf{a} \times \mathbf{b} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$
Angle: $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$, $\theta \in [0, \pi]$
Projection of b onto a: $\text{proj}_\mathbf{a}\mathbf{b} = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|^2}\,\mathbf{a}$, Scalar projection $= \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}$
Scalar Triple Product: $[\mathbf{a},\mathbf{b},\mathbf{c}] = \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \det\begin{pmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{pmatrix}$. Volume of parallelepiped = $|[\mathbf{a},\mathbf{b},\mathbf{c}]|$
Vector Triple Product: $\mathbf{a}\times(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}$ (BAC-CAB rule)