How is the dot product calculated?

The dot product multiplies matching components and adds them up. For a = (1, 2, 3) and b = (4, 5, 6) it is 1·4 + 2·5 + 3·6 = 4 + 10 + 18 = 32. The result is a single number, not a vector. A dot product of zero means the two vectors are perpendicular, which is the fastest way to test for a right angle.

Why does the cross product only work in 3D?

The cross product returns a vector perpendicular to both inputs, and there is only one such direction in 3D space. In 2D there is no third axis to point along, so the calculator reports the signed z-component instead: for (1, 0) and (0, 1) that scalar is 1. For full 3D vectors like (1, 0, 0) and (0, 1, 0) you get the vector (0, 0, 1).

What is the magnitude formula?

Magnitude (length) is the square root of the sum of the squared components. For (3, 4) it is √(3² + 4²) = √25 = 5. For a 3D vector (1, 2, 2) it is √(1 + 4 + 4) = √9 = 3. The tool uses a hypot routine internally so very large or very small components do not overflow or lose precision.

How do you find the angle between two vectors?

The angle comes from the cosine rule cos θ = (a · b) / (|a| · |b|), then θ = arccos of that value, converted to degrees. Perpendicular vectors give 90°, parallel give 0°, and opposite directions give 180°. The cosine is clamped to the range −1 to 1 so rounding never pushes arccos into NaN. A zero vector has no direction, so the angle is undefined and the tool says so.

What is a unit vector and how do I get one?

A unit vector points the same way as the original but has length exactly 1. You get it by dividing each component by the magnitude. Normalising (3, 4) gives (0.6, 0.8), whose length is 1. Unit vectors are how you separate direction from size, which is handy for surface normals, headings and lighting. The zero vector cannot be normalised because its length is 0.

Should I enter 2D or 3D vectors, and what about mismatched sizes?

Enter two or three numbers separated by commas — the tool reads the dimension from how many you type. Add, subtract, dot and angle require both vectors to have the same dimension, so mixing a 2D and a 3D vector is reported as an error rather than silently padded. Magnitude and the unit vector work on either vector on its own. Cross product accepts 2D or 3D as long as both match.

Check physics homework on forces and motion

A statics problem gives two force vectors and asks for the resultant and the angle between them. Type both forces as components, read the sum directly, then switch to the angle operation to confirm θ. When the dot product comes back as 0 you know instantly the two forces are perpendicular, which is often the whole point of the question. No re-deriving the cosine rule by hand under exam pressure.

Compute a surface normal in a graphics or game project

You have two edge vectors of a triangle and need the normal for lighting. Take the cross product of the two edges to get a perpendicular vector, then normalise it to length 1 so the shading math behaves. Paste both edges as 3D components, run cross product, switch to the unit-vector operation, and copy the three numbers straight into your shader uniform.

Decompose a vector for an engineering calculation

Projecting a velocity or force onto a direction is the everyday version of "how much of this points along that". Enter the vector as a and the direction as b, run the projection, and you get both the projected vector and the signed scalar length along b. That scalar is exactly the component you feed into a work-done or along-track distance calculation.

Verify a linear-algebra exercise step by step

Working through a textbook chapter on dot products, norms and angles, you want a quick oracle to check each result before moving on. Punch in the same vectors the book uses, compare the magnitude, dot product and angle, and you catch an arithmetic slip in seconds instead of trusting an answer key that might have a typo. Share the URL with a study partner and the exact problem reopens.

Sanity-check a 2D heading or movement vector in code

Debugging a 2D game where a sprite drifts off course, you suspect the movement vector is not normalised. Drop the raw vector in, read its magnitude, and if it is not 1 you have found the bug. Normalise it in the tool, copy the unit vector, and compare against what your code produces to pin down where the length crept in.

Vector Calculator — Dot, Cross, Magnitude, Angle

Add, subtract, scale, dot, cross, magnitude, angle, unit vector and projection for 2D and 3D vectors — browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Enter two vectors as comma-separated components (2D or 3D). Pick an operation and read the result. Cross product needs 3D; angle and projection need a non-zero second vector. Everything runs in your browser.

Try an example

Vector a

Vector b

Operation

Result

a · b = 32

All vector math runs in JavaScript inside your browser. The numbers you type never leave the page; the shareable link encodes them in the URL only when you choose to share it.

What this tool does

A focused vector calculator for 2D and 3D vectors that does the operations a physics or linear-algebra course actually asks for. Type each vector as comma-separated components and pick an operation: vector addition and subtraction, scalar multiplication, the dot product (a scalar), the cross product (a 3D vector, with the 2D case shown as the signed z-component), magnitude with the square-root-of-sum-of-squares formula, the angle between two vectors in degrees from the cosine rule, unit vectors, and the projection of one vector onto another. A single vector is enough for magnitude and normalisation. The cosine value is clamped to keep the angle stable against floating-point noise, and zero-vector cases are reported instead of returning a misleading NaN. One-click copy and a shareable URL that reopens with your exact vectors and operation. Everything runs in your browser; no numbers are uploaded.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Vector Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Check physics homework on forces and motion
A statics problem gives two force vectors and asks for the resultant and the angle between them. Type both forces as components, read the sum directly, then switch to the angle operation to confirm θ. When the dot product comes back as 0 you know instantly the two forces are perpendicular, which is often the whole point of the question. No re-deriving the cosine rule by hand under exam pressure.
Compute a surface normal in a graphics or game project
You have two edge vectors of a triangle and need the normal for lighting. Take the cross product of the two edges to get a perpendicular vector, then normalise it to length 1 so the shading math behaves. Paste both edges as 3D components, run cross product, switch to the unit-vector operation, and copy the three numbers straight into your shader uniform.
Decompose a vector for an engineering calculation
Projecting a velocity or force onto a direction is the everyday version of "how much of this points along that". Enter the vector as a and the direction as b, run the projection, and you get both the projected vector and the signed scalar length along b. That scalar is exactly the component you feed into a work-done or along-track distance calculation.
Verify a linear-algebra exercise step by step
Working through a textbook chapter on dot products, norms and angles, you want a quick oracle to check each result before moving on. Punch in the same vectors the book uses, compare the magnitude, dot product and angle, and you catch an arithmetic slip in seconds instead of trusting an answer key that might have a typo. Share the URL with a study partner and the exact problem reopens.
Sanity-check a 2D heading or movement vector in code
Debugging a 2D game where a sprite drifts off course, you suspect the movement vector is not normalised. Drop the raw vector in, read its magnitude, and if it is not 1 you have found the bug. Normalise it in the tool, copy the unit vector, and compare against what your code produces to pin down where the length crept in.

Common pitfalls

Confusing the dot product with the cross product. The dot product returns a number and tests for perpendicularity; the cross product returns a vector perpendicular to both. Asking for a cross product when you wanted a scalar similarity, or vice versa, gives a result of the wrong type every time.
Trying to take a cross product of mismatched or higher-dimension vectors. The cross product is a 2D or 3D operation only, and both vectors must share the same dimension. Feeding a 2D vector against a 3D one is reported as an error rather than silently padded with a zero, which would hide a real input mistake.
Expecting an angle or unit vector from a zero vector. A vector of length 0 has no direction, so its unit vector and any angle involving it are undefined. The tool says so instead of returning NaN, but it is easy to forget that an all-zeros input is the cause when a result comes back blank.

Privacy

Every operation — the dot and cross products, the magnitude, the angle, the unit vectors and the projection — is plain JavaScript that runs in your browser tab. The vectors you type never leave the page, and nothing is logged. The one caveat: the shareable URL encodes your two vectors and the chosen operation in the query string, so a "share link" pasted into chat will record those numbers in the recipient server's access log. For anything you would rather keep private, use the copy button and paste the result text instead of sharing the URL.

FAQ

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Vector Calculator — Dot, Cross, Magnitude, Angle

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Vector Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Check physics homework on forces and motion

Compute a surface normal in a graphics or game project

Decompose a vector for an engineering calculation

Verify a linear-algebra exercise step by step

Sanity-check a 2D heading or movement vector in code

Common pitfalls

Privacy

FAQ

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