Vector Calculator

A vector has both a size and a direction, and almost every question you can ask about two vectors comes down to a handful of formulas. This calculator applies all of them to the vectors you enter, in two or three dimensions, and writes out the arithmetic for whichever operation you pick. Whole numbers, decimals and fractions stay exact through every sum and product, so a projection such as (−5/7) · ⟨1, 4, −2⟩ appears as ⟨−5/7, −20/7, 10/7⟩ instead of a rounded approximation.

If your vectors come out of a set of equations rather than out of geometry, the System of Equations Calculator and the Matrix Calculator cover that side of linear algebra. For the straight-line distance between two points, the Distance Calculator is quicker.

What each vector operation tells you

The operations answer different questions. Addition and subtraction combine displacements: moving by ⟨3, −2, 5⟩ and then by ⟨1, 4, −2⟩ leaves you at ⟨4, 2, 3⟩. A scalar multiple stretches or shrinks a vector, and it reverses the direction when the scalar is negative. The magnitude is the vector's length, and the unit vector keeps the direction while scaling the length to exactly 1.

The two products raise most of the questions. The dot product is a single number that measures how strongly two vectors point the same way, which is why it leads straight to the angle between them. The cross product is a new vector at right angles to both inputs, and its length equals the area of the parallelogram they span. The projection splits one vector into a part along another vector and a part perpendicular to it, and the triple scalar product of three vectors gives the volume of the slanted box they form.

The vector formulas

With a = ⟨a₁, a₂, a₃⟩, b = ⟨b₁, b₂, b₃⟩ and c = ⟨c₁, c₂, c₃⟩, these are the definitions the calculator uses:

For 2D vectors, drop every third component. The cross product then treats both vectors as having a z component of 0, which leaves only the last entry, a₁b₂ − a₂b₁. Two identities tie the products together and make excellent checks: |a × b| = |a| |b| sin θ, and Lagrange's identity, |a × b|² = |a|²|b|² − (a · b)².

Worked example 1: dot product and angle between two 3D vectors

Take a = ⟨3, −2, 5⟩ and b = ⟨1, 4, −2⟩, the default vectors in the calculator. Multiply matching components and add:

The negative sign already tells you the angle is wider than 90°. To pin it down you need both lengths:

Keep the square roots exact until the final step. If you round the lengths to 6.16 and 4.58 first, the cosine becomes −0.5317 and the angle comes out as 122.1186°, about 0.05° off. Rounding them to 6.2 and 4.6 gives 121.7320°, a third of a degree away, which is enough to miss an answer key that expects four decimals.

Worked example 2: the cross product by determinant expansion

For the same two vectors, put the unit vectors i, j and k in the top row of a determinant, a in the middle row and b in the bottom row, then expand along the top row. The middle term carries a minus sign, which is the step most often lost by hand:

Two quick checks confirm the result. First, the cross product must be perpendicular to both inputs: ⟨−16, 11, 14⟩ · ⟨3, −2, 5⟩ = −48 − 22 + 70 = 0, and ⟨−16, 11, 14⟩ · ⟨1, 4, −2⟩ = −16 + 44 − 28 = 0. Second, Lagrange's identity says the squared length must be 38 × 21 − (−15)² = 798 − 225 = 573. The components give 256 + 121 + 196 = 573, so |a × b| = √573 ≈ 23.9374. That is the area of the parallelogram spanned by a and b; the triangle with the same two sides has half of it, about 11.9687.

Add a third vector c = ⟨2, 0, 1⟩ and the triple scalar product a · (b × c) is the 3 × 3 determinant with rows a, b and c: 3(4 − 0) − (−2)(1 + 4) + 5(0 − 8) = 12 + 10 − 40 = −18. The box with edges a, b and c therefore has a volume of 18, the tetrahedron on the same edges has a volume of 3, and the negative sign means the three vectors form a left-handed set.

Worked example 3: unit vector, angle and projection in 2D

Switch to 2D and enter a = ⟨7, 1⟩ and b = ⟨3, 4⟩, or tap the 2D preset. The lengths are |a| = √50 = 5√2 ≈ 7.0711 and |b| = √25 = 5, so the unit vector of b is exact: b̂ = ⟨3/5, 4/5⟩ = ⟨0.6, 0.8⟩. The dot product is 21 + 4 = 25, which gives a clean angle:

The projection of a onto b is (25/25) · ⟨3, 4⟩ = ⟨3, 4⟩, so the shadow of a on b happens to be b itself, with a scalar projection of 25/5 = 5. Subtracting it leaves the perpendicular part ⟨4, −3⟩, and ⟨4, −3⟩ · ⟨3, 4⟩ = 0 confirms the split. Treating both vectors as 3D vectors with z = 0 gives a × b = ⟨0, 0, 25⟩: the parallelogram has an area of 25, and the positive sign means b lies counterclockwise from a. The calculator also reports the direction of each 2D vector measured from the positive x-axis, 8.1301° for a and 53.1301° for b, and their difference is the same 45°.

Reference table: 3D vectors with whole-number length

Most vectors have irrational lengths, which is why textbooks lean on a small family whose components form Pythagorean quadruples. Their unit vectors are exact fractions, and those fractions are also the direction cosines, the cosines of the angles α, β and γ that the vector makes with the x-, y- and z-axes. Every value below was computed with the same code the calculator runs.

Two patterns are worth noticing. Scaling a vector never changes its unit vector, so ⟨2, 4, 4⟩ shares the unit vector ⟨1/3, 2/3, 2/3⟩ with ⟨1, 2, 2⟩. And the squares of the three direction cosines always add up to 1: for ⟨2, 3, 6⟩, (2/7)² + (3/7)² + (6/7)² = 4/49 + 9/49 + 36/49 = 1.

Dot product, cross product and angle for common vector pairs

The table pairs vectors that come up again and again in homework and geometry. Angles are rounded to four decimals; the exact ones are marked.

The cube rows are a classic: the angle between an edge of a cube and its space diagonal is arccos(1/√3) ≈ 54.7356°, and the angle between a face diagonal and the space diagonal is arccos(2/√6) ≈ 35.2644°. The opposite pair shows a zero cross product at 180°, and both orthogonal pairs show a zero dot product at 90°.

How to use this calculator

Choose 2D or 3D, then type the components of a and b. Each box accepts whole numbers, decimals and fractions such as 3/4 or −1/2. The scalar k feeds the scalar multiple, and vector c, which appears only in 3D mode, feeds the triple scalar product. The Swap button exchanges a and b, which makes it easy to compare a × b with b × a, or the projection of a onto b with the projection of b onto a.

Pick an operation to see its result in the blue card and its full working in the step-by-step card below. The parallel or orthogonal card runs both tests, and the table at the bottom lists every quantity at once. The decimal places buttons control rounding only; exact fractions and radicals such as √38 are never rounded. A decimal too small for the chosen number of places, or 10¹⁵ and above, is shown in e-notation, such as 1.414e−10, rather than as 0 or a long row of digits.

Who uses vector operations

Students meet these operations in precalculus, multivariable calculus, linear algebra and physics, usually as homework that asks for a cross product or an angle with every step written out. Physics and engineering rely on them constantly. The work done by a force is the dot product W = F · d, so a force of ⟨20, 15, 0⟩ newtons that moves an object by ⟨30, 10, 0⟩ meters does 600 + 150 = 750 joules of work. Torque is the cross product τ = r × F, so a 40-newton push along the y-axis applied 0.3 meters out along the x-axis gives ⟨0, 0, 12⟩ newton meters about the z-axis.

Computer graphics programmers use the cross product to build surface normals and the dot product to shade a surface by the angle between its normal and the light. Game and mapping software uses the sign of the 2D cross product to decide which side of a line a point lies on. The cross product also measures triangles in space: for the points P(1, 0, 2), Q(4, 2, 3) and R(2, 5, 1), the edge vectors PQ = ⟨3, 2, 1⟩ and PR = ⟨1, 5, −1⟩ have cross product ⟨−7, 4, 13⟩ with length √234 ≈ 15.2971, so the triangle's area is about 7.6485.

Key terms

• Vector: a quantity with magnitude and direction, written in components such as ⟨3, −2, 5⟩.
• Scalar: an ordinary number. The dot product and the triple product return scalars.
• Magnitude (norm, length): |a| = √(a · a), never negative.
• Unit vector: a vector of length 1; â = a / |a| points the same way as a.
• Dot product (scalar or inner product): a · b = a₁b₁ + a₂b₂ + a₃b₃.
• Cross product (vector product): a × b, perpendicular to a and b, with length |a| |b| sin θ.
• Orthogonal: perpendicular; the dot product is 0.
• Parallel: one vector is a scalar multiple of the other; the cross product is the zero vector.
• Projection: the part of a that lies along b, as a vector (proj_b a) or as a signed length (the scalar projection).
• Right-hand rule: curl the fingers of your right hand from a toward b and your thumb points along a × b.
• Triple scalar product: a · (b × c), the signed volume of the box with edges a, b and c.

What changes the result, and how to check it

• Order in the cross product. b × a = −(a × b), so swapping the vectors flips every component: for the worked example, b × a = ⟨16, −11, −14⟩. The dot product and the angle do not depend on order.
• Direction of a projection. The projection of a onto b and the projection of b onto a are different vectors. In the 2D example they are ⟨3, 4⟩ and ⟨7/2, 1/2⟩.
• Dimension. A 2D vector is the same as a 3D vector with z = 0. A stray value in the z box changes the magnitude, the angle and the cross product.
• Angle units. Most software returns arccos in radians. The values 2.1306 rad and 122.0727° describe the same angle; multiply radians by 180/π to convert.
• Zero vectors. A zero vector has no direction, so its unit vector, its angle with another vector and any projection onto it are undefined.

Three checks catch most hand-calculation errors. The cross product must have a dot product of 0 with both inputs. Its squared length must equal |a|²|b|² − (a · b)². And the angle must agree with the signs: a positive dot product means an acute angle, a negative one an obtuse angle.

Limitations

The calculator handles vectors with two or three numeric components. It does not accept variables, so it cannot simplify the cross product of symbolic vectors, and it does not work with complex components or with four or more dimensions, where the familiar cross product is not defined. Square roots, angles, unit vectors and scalar projections are displayed as decimals rounded to the precision you choose, so read a displayed 0.7071 as √2/2 rather than as an exact value. Large inputs are still computed exactly, but a square root is only simplified into radical form while the number under it stays below about 10¹⁵; beyond that you get the decimal value alone.

Related calculators

• To solve two, three or four linear equations at once, use the System of Equations Calculator.
• For determinants, inverses and matrix products, use the Matrix Calculator.
• To row-reduce a matrix with every row operation shown, use the RREF Calculator.
• To solve a single equation for x with steps, use the Algebra Calculator or, for ax + b = c, the Linear Equation Calculator.
• For the straight-line distance between two points, use the Distance Calculator.
• For sine, cosine and angle conversions, use the Trigonometry Calculator.

Sources

Vector algebra is deterministic mathematics, so every figure on this page follows from the definitions rather than from published data. The definitions and identities used here match these references:

• NIST Digital Library of Mathematical Functions, §1.6 Vectors and Vector-Valued Functions: the dot product (1.6.2), the norm (1.6.3), the angle between vectors (1.6.4), the cross product as a determinant (1.6.9) and the parallelepiped volume |a · (b × c)|.
• MIT OpenCourseWare 18.02SC Multivariable Calculus, Notes: Cross Product: the determinant definition, |A × B| = |A||B| sin θ as the parallelogram area, the right-hand rule, Lagrange's identity and the triangle-area method.
• MIT OpenCourseWare 18.02SC, Session 2: Dot Products: the component formula and its link to lengths and angles.
• NIST Special Publication 811, Guide for the Use of the International System of Units: the radian as the SI unit of plane angle and the degree as π/180 rad.
• All worked examples and table values were computed with the calculator's own exact-fraction code and cross-checked in floating point.