Dot Product Calculator

1. Enter the components of Vector A, separated by commas.
2. Enter Vector B with the same number of components.
3. Read the dot product, a single number.
4. See the angle between the vectors and whether they are perpendicular.

Here is what each result means:

The dot product, also called the scalar product, combines two vectors into a single number. You multiply matching components and add the results. So for the vectors 1, 2, 3 and 4, 5, 6, the dot product is 1 times 4 plus 2 times 5 plus 3 times 6, which is 32.

The dot product measures how much two vectors point the same way. It equals the product of their lengths times the cosine of the angle between them, so a positive value means a sharp angle, zero means a right angle, and a negative value means an obtuse angle. That link makes it central to geometry, physics and graphics.

It multiplies matching components and adds them, then uses the lengths to find the angle.

1. Multiply each pair of matching components.
2. Add those products for the dot product.
3. Divide by the two magnitudes and take the inverse cosine for the angle.

The lengths in the angle formula come from the vector magnitude calculator.

Find the dot product of 1, 2, 3 and 4, 5, 6.

Calculation: 1 x 4 + 2 x 5 + 3 x 6 = 4 + 10 + 18 = 32. The result is positive, so the angle is sharp; working out the lengths gives an angle of about 12.9 degrees between the two vectors.

The sign alone tells you the rough angle between the vectors.

A dot product of zero is the quick test for two vectors being at right angles, without finding the full angle.

The two vector products give very different things.

The dot product works in any dimension and gives a scalar; the cross product is defined in three dimensions and gives a vector.

The Angle Between the Vectors

The dot product is largest when the vectors point the same way and zero when they are perpendicular.

The Lengths of the Vectors

Longer vectors give a larger dot product for the same angle, since it scales with both lengths.

The Sign of Components

Opposing components pull the total down and can make the dot product negative.

Finding an Angle

Work out the angle between two vectors from their components.

Testing Perpendicularity

Check whether two vectors are at right angles with a single number.

Physics

Compute work as force dotted with displacement, or a projection onto a direction.

1. Expecting a Vector

The dot product is a single number, not a vector. The cross product gives a vector.

2. Mismatched Lengths

Both vectors must have the same number of components to pair them up.

3. Forgetting to Add

Multiply matching components and then add all the products. A single product is not the dot product.

4. Misreading a Negative Result

A negative dot product is valid and means an obtuse angle, not an error.

5. Confusing It with the Cross Product

They are different operations with different results and uses.

The dot product is exact; only the displayed decimals are rounded.

It is a single number found by multiplying two vectors term by term and adding the results. For 1, 2, 3 and 4, 5, 6 it is 4 plus 10 plus 18, which is 32. It measures how aligned the vectors are.

Multiply each pair of matching components, then add all the products. Both vectors must have the same number of components. This tool does that and also finds the angle between them.

It shows how much two vectors point the same way. A positive value means a sharp angle, zero means perpendicular, and a negative value means an obtuse angle beyond 90 degrees.

When the two vectors are perpendicular, at exactly 90 degrees. So a dot product of zero is a quick test for two vectors being at right angles.

Divide the dot product by the product of the two lengths, then take the inverse cosine. This tool reports the angle in degrees for you.

The dot product gives a single number that measures alignment and works in any dimension. The cross product gives a new vector perpendicular to both and is defined in three dimensions.

Yes. A negative dot product means the angle between the vectors is greater than 90 degrees, so they point in broadly opposite directions. It is a valid result.

It finds angles, tests perpendicularity, and projects one vector onto another. In physics it computes work as force dotted with displacement.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Dot product (Wikipedia).
• The dot product explained (Maths Is Fun).
• Vectors (Wikipedia).