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Free Online Vector Calculator With Dot and Cross Product

Add, subtract, scale, dot and cross two vectors in two or three dimensions, and get the length of each, the angle between them and the projection of one onto the other. The angle stays accurate for nearly parallel vectors, where the usual arc cosine formula collapses, and perpendicular and parallel pairs are named.

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Free Online Vector Calculator With Dot and Cross Product

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➹ Vector Calculator Two or three dimensions · dot, cross and angle · accurate on near parallel pairs

The sum runs in your browser. Nothing you type is sent anywhere.

What to work out

➹ The vectors

Two or three numbers, separated by commas or spaces. Brackets are ignored, so (3, -2, 5) and 3 -2 5 both work.

Vector a

Vector b

Multiply every part by

Try one of these

The fourth is where the usual angle formula gives zero or an error.

The answer

—

Length of a—

Length of b—

Dot product—

Angle between them—

How they sit—

Each result with your own numbers put into the formula.

What	Formula	With your numbers

Recent vector pairs, kept in this browser only.

Nothing here yet Press Log to history on the Calculate tab to keep a pair here.

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How to Use the Vector Calculator

Three steps, and the third is the interesting one.

Choose what to work out Add, subtract, scale, dot product, cross product, the angle between them, a unit vector or a projection. Only the boxes that operation needs are shown.

Type the vectors Two or three numbers separated by commas or spaces. A two dimensional vector is treated as a three dimensional one with a zero on the end where that is needed, such as for a cross product.

Read the answer and the rows beneath The lengths, the dot product and the angle are given whatever operation you chose, because they are what tell you how the two vectors sit relative to each other.

What to Know About Vector Arithmetic

Dot against cross, and where the angle formula fails.

The angle survives nearly parallel vectors. The textbook formula is the arc cosine of the dot product over the product of the lengths. For two vectors pointing almost the same way that fraction rounds to exactly one, and the arc cosine of a value a shade above one is not a number at all. Here the cross product form is used in that region instead, which is accurate precisely where the other one is not: the angle between (1, 0, 0) and (1, one billionth, 0) comes out as a real, tiny angle rather than zero or an error.

The dot product is a number, the cross product is a vector. This is the thing most often mixed up. The dot product measures how much two vectors point the same way, and it is zero exactly when they are perpendicular. The cross product produces a new vector perpendicular to both, whose length is the area of the parallelogram the two of them span. They answer different questions and are not interchangeable.

The cross product changes sign when you swap the vectors. a cross b and b cross a point in opposite directions. The dot product, by contrast, is the same either way. The swap button makes this easy to see, and it matters whenever a cross product is used for an orientation or a surface normal.

Two dimensional vectors get a zero third part where they need one. A cross product only exists in three dimensions, so a pair of flat vectors is treated as lying in the z equals zero plane. The result then points straight up or down, and its third part is the signed area of the parallelogram, which is the useful two dimensional quantity.

The zero vector has no direction. It has no unit vector, and no angle can be measured to it, because there is nothing to point. Rather than dividing by a length of zero and returning not a number, the page says so.

It runs in your browser with nothing uploaded. The arithmetic is in the page, so it works offline once loaded and nothing you type is sent anywhere.

Key Features & Capabilities

What it does, and what it deliberately does not.

Eight operations Add, subtract, scale, dot, cross, angle, unit vector and projection, in two or three dimensions.

An angle that stays accurate The cross product form takes over where the arc cosine formula loses its digits.

Lengths and dot product always shown Whatever you asked for, the rows beneath tell you how the two vectors sit relative to each other.

Perpendicular and parallel named Because a dot product of zero and an angle of ninety degrees are the same fact said twice.

The working shown Your own numbers put into each formula, so it can be copied as a method.

Nothing is uploaded It runs in this tab and keeps working offline.

About Vectors

A vector is a quantity with both a size and a direction, written as a list of components. Two of them can be added, which puts them nose to tail, or scaled, which stretches one without turning it. Those operations are straightforward. The two products are where the ideas live.

The dot product multiplies matching components and adds the results, giving a single number. That number is the product of the two lengths times the cosine of the angle between them, which means it is positive when they point broadly the same way, negative when they point apart, and exactly zero when they are perpendicular. That last fact is the most used property in all of vector algebra: testing for a right angle is a multiplication and an addition, with no trigonometry at all.

The cross product exists only in three dimensions and produces a vector rather than a number. It points perpendicular to both inputs, and its length is the product of the two lengths times the sine of the angle, which is also the area of the parallelogram the two vectors span. Swapping the inputs reverses it, which is why it is used to define orientation, torque and surface normals.

The angle between two vectors is where a calculator can quietly go wrong. Rearranging the dot product gives the angle as an arc cosine, and that works for most pairs. For two vectors pointing almost the same way the fraction inside rounds to one, and the arc cosine then returns zero for any of a wide range of genuinely different small angles, or fails outright when rounding pushes the fraction a shade above one. The alternative is to compute the angle from the cross product and the dot product together, which keeps its accuracy exactly where the first form loses it. Both are used here, each in the region where it is the better one.

Frequently Asked Questions

Dimensions, products and what the results mean.

The dot product gives a single number that measures how much the two vectors point the same way, and it is zero when they are perpendicular. The cross product gives a new vector perpendicular to both, whose length is the area of the parallelogram they span.

Yes. Everything works in two dimensions, and for a cross product the pair is treated as lying flat in three dimensional space, so the result points straight up or down and its size is the signed area.

Because a cross b and b cross a point in opposite directions. The dot product is unchanged by swapping; the cross product changes sign. That is a property of the operation rather than an error.

Their dot product is zero. The page says so directly in the row beneath the answer, so there is no need to work out the angle first.

The part of the first vector that lies along the second: the shadow it would cast on that direction. Subtracting it from the original leaves the part that is perpendicular, which is how vectors are split into components.

Because it has no direction to measure from. Dividing by its length of zero would give not a number, so the page says plainly that there is no angle rather than printing one.

No. Every operation runs in your browser, so nothing you type leaves the page and it works offline.