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Walk-through

How to Use This Calculator

3 steps

1

Choose 2D or 3D and enter both vectors

Pick 2D or 3D from the dropdown, then enter the x, y (and z, if 3D) components of vector A and vector B. Add an optional scalar k if you want to see A scaled by that value — every field recalculates instantly as you type.

2

Switch tabs to see each operation

The Add/Subtract tab shows A + B and A − B side by side (plus the scaled vector). The Dot/Cross tab shows the dot product and cross product, and flags whether A and B are orthogonal. The Angle & Magnitude tab shows |A|, |B|, and the angle between the two vectors.

3

Try the example chips

Use the example chips above the inputs to load a simple addition, a pair of orthogonal vectors (90° apart), or a 3D pair — a fast way to see how each operation behaves before plugging in your own numbers.

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Reference

Formula & Methodology

5 formulas

Addition, subtraction, and scaling

A ± B = (Ax±Bx, Ay±By, Az±Bz); kA = (kAx, kAy, kAz)

Vectors add and subtract component-by-component: each coordinate of the result comes only from the matching coordinate of A and B. Scaling a vector by k multiplies every component by k — it stretches (|k|>1), shrinks (|k|<1), or reverses (k<0) the vector without changing its direction (unless k is negative).

Magnitude

|v| = √(x² + y² + z²)

The magnitude (length) of a vector is the square root of the sum of its squared components — the Pythagorean theorem extended to 3D. In 2D, z is simply 0 and drops out of the formula.

Dot product

A · B = ΣaᵢBᵢ = AxBx + AyBy + AzBz

The dot product multiplies matching components and sums the results, producing a single scalar (not a vector). It's 0 exactly when A and B are orthogonal (perpendicular), positive when they point in a broadly similar direction, and negative when they point in broadly opposite directions.

Cross product

A × B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx)

In 3D, the cross product returns a new vector perpendicular to both A and B, with magnitude equal to the area of the parallelogram they span. In 2D, only the z-component of this formula survives (since Az = Bz = 0), so the '2D cross product' is the single scalar AxBy − AyBx — useful for signed area and orientation (clockwise vs. counterclockwise).

Angle between two vectors

θ = acos((A · B) / (|A| · |B|))

Rearranging the dot-product formula A·B = |A||B|cos(θ) gives the angle directly. This is undefined when either vector has zero magnitude, since there's no defined direction to measure an angle from.

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Glossary

Key Terms Explained

7 terms

Vector A quantity with both magnitude (size) and direction, represented here as an ordered list of components — (x, y) in 2D or (x, y, z) in 3D.

Magnitude The length of a vector, written |v|, computed as the square root of the sum of its squared components. Always a non-negative number.

Dot product A scalar (single number) formed by multiplying matching components of two vectors and summing the results. Also called the 'scalar product.' Zero exactly when the vectors are orthogonal.

Cross product In 3D, a new vector perpendicular to both input vectors, with magnitude equal to the area of the parallelogram they form. Also called the 'vector product.' Only defined in 3D in its full vector form.

Unit vector A vector with magnitude exactly 1, often used to represent pure direction without scale. Any non-zero vector v can be converted to a unit vector by dividing each component by |v|.

Scalar A single real number, as opposed to a vector. Multiplying a vector by a scalar (scalar multiplication) scales its length and can flip its direction if the scalar is negative.

Orthogonal Two vectors are orthogonal when they meet at a 90° angle — equivalently, when their dot product equals 0. 'Orthogonal' generalizes 'perpendicular' to any number of dimensions.

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Scenarios

Real-World Examples

3 worked examples

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Simple 2D addition

A = (1, 2), B = (3, 4)

Vector A (1, 2) Vector B (3, 4)

Add each pair of matching components: A + B = (1+3, 2+4) = (4, 6). Subtraction works the same way component-by-component: A − B = (1−3, 2−4) = (−2, −2). This is the most basic vector operation — every other operation on this page builds on the same 'component-by-component' idea.

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Orthogonal vectors

A = (1, 0), B = (0, 1)

Vector A (1, 0) Vector B (0, 1)

The dot product is A · B = (1)(0) + (0)(1) = 0. Whenever the dot product is exactly 0, the two vectors are orthogonal — here, A points straight along the x-axis and B points straight along the y-axis, so they meet at a right angle. The Angle & Magnitude tab confirms this: the angle between them is exactly 90°.

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Checking the angle formula

Same orthogonal pair, reading the angle directly

Vector A (1, 0) Vector B (0, 1)

θ = acos((A · B) / (|A| · |B|)) = acos(0 / (1 × 1)) = acos(0) = 90°. Because both A and B are already unit vectors (magnitude 1) in this example, the formula simplifies to just acos(dot product) — a useful mental shortcut whenever you're comparing directions rather than full vectors.

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Deep Dive

Understanding the Vector Calculator

3 chapters · 4 min read

Vectors describe quantities that have both a size and a direction — velocity, force, and displacement are all classic examples. This calculator takes two vectors, A and B, in 2D or 3D, and computes every common operation on them at once: addition, subtraction, scalar multiplication, the dot product, the cross product, magnitude, and the angle between them.

How the Vector Calculator works

Every operation on this page works directly on the vectors' components. Addition and subtraction combine matching components (x with x, y with y, z with z); scalar multiplication multiplies every component by the same number k. The dot product multiplies matching components and adds the results into a single number, while the cross product (3D only) combines components in a specific pattern to produce a new vector perpendicular to both inputs.

Magnitude and angle are both derived from these building blocks: magnitude is the Pythagorean length of a vector's components, and the angle between two vectors falls directly out of rearranging the dot-product formula.

Inputs and what they mean

Vector A and vector B are entered as their x, y (and z, in 3D mode) components — not as a magnitude-and-direction pair. Switching between 2D and 3D changes whether the z-component fields are shown and whether the cross product returns a full 3D vector or a single signed 2D scalar. The scalar k is only used for the 'A scaled by k' result on the Add/Subtract tab; it doesn't affect any of the other computed values.

Limits and edge cases

The angle between two vectors is undefined when either vector has zero magnitude — there's no direction to measure from a point with no length, so the calculator flags this case explicitly rather than showing a misleading number. The cross product's full vector form only exists in 3D; in 2D, what's usually called the 'cross product' is really the z-component of that same formula, reduced to a single signed number that indicates orientation (clockwise vs. counterclockwise) and doubles as the signed area of the parallelogram the two vectors span.

This calculator works with any consistent unit for the vector components (meters, feet, unitless coordinates, etc.) — just use the same unit for every component you enter.

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Questions

Frequently Asked Questions

6 questions

What is the dot product formula?+

The dot product of A = (Ax, Ay, Az) and B = (Bx, By, Bz) is A · B = AxBx + AyBy + AzBz — multiply matching components and add the results. The output is always a single number (a scalar), not a vector.

What does the cross product mean?+

In 3D, the cross product A × B produces a new vector that is perpendicular to both A and B, with a length equal to the area of the parallelogram the two vectors span. In 2D, only the resulting z-component survives, giving a single signed number instead of a full vector.

How is magnitude calculated?+

Magnitude is |v| = √(x² + y² + z²) — the square root of the sum of the squared components. It's the vector's length, and it's always zero or positive.

How is the angle between two vectors calculated?+

θ = acos((A · B) / (|A| · |B|)) — take the dot product, divide by the product of the two magnitudes, then take the inverse cosine. This is undefined if either vector has zero magnitude.

What is the 2D cross product exactly?+

In 2D there's no true cross product in the 3D sense (a vector perpendicular to a plane needs a third dimension to point into). What's commonly called the '2D cross product,' AxBy − AyBx, is really just the z-component of the full 3D formula when both vectors have z = 0. It returns a single signed scalar, not a vector.

What does it mean for two vectors to be orthogonal?+

Two vectors are orthogonal (perpendicular) exactly when their dot product equals 0 — this calculator checks that condition automatically and flags it on the Dot/Cross tab. 'Orthogonal' is the general term that applies in any number of dimensions, while 'perpendicular' is more commonly used in 2D and 3D geometry.

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Vector Calculator

Enter vectors A and B

Resultant Vector

Vector diagram (XY plane)

Dot / Cross Product

Angle Between Vectors

How to Use This Calculator

Choose 2D or 3D and enter both vectors

Switch tabs to see each operation

Try the example chips

Formula & Methodology

Key Terms Explained

Real-World Examples

Simple 2D addition

Orthogonal vectors

Checking the angle formula

Understanding the Vector Calculator

How the Vector Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Vector Calculator

Enter vectors A and B

Resultant Vector

Vector diagram (XY plane)

Dot / Cross Product

Angle Between Vectors

How to Use This Calculator

Choose 2D or 3D and enter both vectors

Switch tabs to see each operation

Try the example chips

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Vector Calculator

How the Vector Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

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