Vectors & Matrices formulas — 24 solvers

Angle Between Two 2D Vectors (Components)

\theta = \arccos\left(\frac{a_x b_x + a_y b_y}{\sqrt{a_x^2+a_y^2}\,\sqrt{b_x^2+b_y^2}}\right)

Vectors & MatricesTrigonometryAlgebraFinds the angle separating two plane vectors directly from their four components, by way of the normalized dot product.

Cross Product Magnitude

|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta

Vectors & MatricesTrigonometryPhysicsGives the length of the cross product of two vectors from their magnitudes and the angle between them, equal to the area they span.

Cross Product z-Component of Two 2D Vectors

(\vec{a}\times\vec{b})_z = a_x b_y - a_y b_x

Vectors & MatricesAlgebraGeometryComputes the signed out-of-plane cross product of two plane vectors from their components, whose sign reveals their turning direction.

Determinant of a 2×2 Matrix

D = ad - bc

Vectors & MatricesAlgebraComputes the determinant of a two-by-two matrix as the difference of its diagonal products, the number that decides invertibility.

Determinant of a 3×3 Matrix

D = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

Vectors & MatricesAlgebraComputes the determinant of a three-by-three matrix by cofactor expansion along the first row, using all nine entries.

Direction Angle of a 2D Vector

\theta = \operatorname{atan2}(v_y,\; v_x)

Vectors & MatricesTrigonometryAlgebraGives the direction a two-dimensional vector points, measured counterclockwise from the positive x-axis, from its two components.

Dot Product from Magnitudes and Included Angle

\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta

Vectors & MatricesTrigonometryPhysicsGives the scalar product of two vectors from their lengths and the angle between them, the geometric face of the dot product.

Dot Product of Two 2D Vectors (Components)

\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y

Vectors & MatricesAlgebraMultiplies two plane vectors component by component and adds the results, giving the scalar that measures how much they share a direction.

Dot Product of Two 3D Vectors (Components)

\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z

Vectors & MatricesAlgebraMultiplies two space vectors component by component and sums the three products, the standard test for perpendicularity in three dimensions.

Inverse of a 2×2 Matrix (Top-Left Element)

(M^{-1})_{11} = \frac{d}{ad - bc}

Vectors & MatricesAlgebraGives the top-left entry of the inverse of a two-by-two matrix: the opposite diagonal entry divided by the determinant.

Magnitude of a 2D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2}

Vectors & MatricesAlgebraGeometryFinds the length of a two-dimensional vector from its x- and y-components, the Pythagorean theorem written for arrows instead of triangles.

Magnitude of a 3D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Vectors & MatricesAlgebraGeometryFinds the length of a three-dimensional vector from its x-, y-, and z-components, extending Pythagoras into space.

Magnitude of a Scaled Vector

|k\vec{v}| = |k|\,|\vec{v}|

Vectors & MatricesAlgebraGives the length of a vector after multiplying it by a scalar, which stretches or shrinks the arrow without ever bending it.

Parallelogram Area from Two Vectors

A = \left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the parallelogram spanned by two plane vectors directly from their four components, no angle or height required.

Resultant of Two Vectors at an Angle

R = \sqrt{A^2 + B^2 + 2AB\cos\theta}

Vectors & MatricesTrigonometryPhysicsFinds the magnitude of the sum of two vectors from their lengths and the angle between them, the parallelogram rule in one equation.

Scalar Projection of One Vector onto Another

\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Vectors & MatricesAlgebraPhysicsMeasures how far a vector reaches along the direction of another, the length of the shadow it casts on that second vector.

Scalar Triple Product (Parallelepiped Volume)

V = \left|\vec{a}\cdot(\vec{b}\times\vec{c})\right|

Vectors & MatricesGeometryAlgebraFinds the volume of the parallelepiped spanned by three space vectors, from the nine components, as the size of their scalar triple product.

Trace of a 2×2 Matrix

\operatorname{tr}(M) = a_{11} + a_{22}

Vectors & MatricesAlgebraAdds the two entries on the main diagonal of a square matrix, a quantity unchanged by any change of basis.

Triangle Area from Two Vectors

A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the triangle formed by two plane vectors from a shared vertex, using only their components — the shoelace formula.

Unit Vector Component (Normalization)

\hat{u}_x = \frac{v_x}{|\vec{v}|}

Vectors & MatricesAlgebraScales one component of a vector down to the matching component of the unit vector that points the same way, by dividing by the magnitude.

Vector Subtraction (Component Form)

c_x = a_x - b_x

Vectors & MatricesAlgebraPhysicsSubtracts one vector from another one component at a time, the operation behind every relative-velocity and displacement-change problem.

Work from Force and Displacement Components

W = F_x d_x + F_y d_y

Vectors & MatricesPhysicsMechanicsComputes the work done by a force from the components of the force and the displacement, without needing the angle between them.

x-Component from Magnitude and Angle

v_x = |\vec{v}| \cos\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its horizontal part from the vector's length and the angle it makes with the positive x-axis.

y-Component from Magnitude and Angle

v_y = |\vec{v}| \sin\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its vertical part from the vector's length and the angle it makes with the positive x-axis.