Dot Product Calculator

Multiply matching vector components, add them, and use the scalar result to study angle or projection.

Drag & drop or upload an image or PDF

Enter two vectors, such as dot([1,2,2],[2,1,-2])

Enter vectors in brackets, such as [1, 2, 3]. Keep both vectors in the same coordinate order.

Try an example

The dot product turns two vectors into a scalar

In component form, multiply corresponding entries and add the products. The vectors must have the same dimension, and the result is one number rather than another vector.

The geometric form connects that scalar to the angle between nonzero vectors. A zero dot product means two nonzero real vectors are perpendicular, while projection uses the dot product to measure how much one vector points along another.

How to use the dot product calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Component form of the dot product

Multiply components in matching positions and add those products to obtain a scalar.

a \cdot b = i = 1 \sum n a_{i} b_{i}

A reliable way to work through it

Check equal dimensions

Both vectors need the same number of components in the same coordinate order.

Multiply and add

Write every component product, keep its sign, and sum the results once.

Interpret only when defined

Angle and projection formulas that divide by a magnitude require the relevant vectors to be nonzero.

Worked example

Find the dot product and angle

a = 122, b = 21 - 2

Both vectors have three components.

a \cdot b = 1 (2) + 2 (1) + 2 (- 2) = 0

Multiply corresponding components and add.

cos θ = \frac{0}{∥ a ∥∥ b ∥} = 0

Both vectors are nonzero, so the angle formula applies.

θ = 9 0^{\circ}

A cosine of zero gives a right angle.

The dot product is 0, so the two nonzero vectors are perpendicular.

Common mistakes to check

Using different vector dimensions

The component dot product requires the same number of entries in both vectors.

Returning a vector

A dot product is a scalar. A perpendicular vector is associated with the cross product.

Dividing by a zero magnitude

The zero vector has no direction, so its angle and normalized projection formulas are undefined.

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Questions students ask

Is the dot product a vector?

No. The result is a scalar.

What does a zero dot product mean?

For two nonzero real vectors, it means they are perpendicular. If one vector is zero, the angle is undefined.

How do I find the angle between vectors?

Divide the dot product by the product of the magnitudes, then take arccos.

Can a dot product be negative?

Yes. For nonzero real vectors, a negative dot product corresponds to an obtuse angle.