Vector Algebra Calculator

Compute dot product, cross product, magnitude, unit vector, angle between vectors, projections, and triple products — with complete step-by-step working.

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Dimension

Key Formulas

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)

Vector algebra: formulas used

For vectors a and b: the dot product is a·b = |a||b| cos θ = a1b1 + a2b2 + a3b3; the angle is θ = arccos(a·b / (|a||b|)). The cross product a × b (defined in 3D) is perpendicular to both vectors and its magnitude |a||b| sin θ equals the area of the parallelogram they span. The unit vector is a / |a|. Two vectors are perpendicular when a·b = 0 and parallel when a × b = 0.

Common questions

What is the difference between dot and cross product?

The dot product is a scalar measuring alignment; the cross product is a vector perpendicular to both inputs.

Is the cross product commutative?

No. a × b = −(b × a).

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