Formula solver index — formula.expert

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.