Vector Dot Product Calculator

1. What is the dot product?

A vector represents a quantity with both magnitude and direction, such as force, velocity, or displacement. The dot product (also called the scalar product) is a way to multiply two vectors and obtain a single number that measures how aligned they are.

For two 3D vectors A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z), the dot product is defined component-wise as

Dot product formula (components):

A · B = A_xB_x + A_yB_y + A_zB_z

Geometrically, the dot product can also be written in terms of lengths and the angle θ between the vectors:

Here |A| and |B| are the magnitudes (lengths) of the vectors. This relationship is the key to finding the angle between two vectors from their components.

2. Vector magnitudes and the angle between two vectors

The magnitude (length) of a 3D vector is found using the 3D version of the Pythagorean theorem:

Magnitude of A: |A| = √(A_x² + A_y² + A_z²)

Magnitude of B: |B| = √(B_x² + B_y² + B_z²)

Once you know A · B, |A|, and |B|, you can solve for the angle θ between the vectors using the geometric dot product formula rearranged as:

Angle formula:

The calculator computes θ in degrees. If either vector has zero magnitude, the angle between them is undefined because you cannot determine a direction for a zero vector.

3. Interpreting the dot product result

The sign and size of the dot product give quick geometric information:

• Positive dot product (A · B > 0): the angle θ is acute (between 0° and 90°). The vectors point mostly in the same general direction.
• Zero dot product (A · B = 0): the angle θ is 90°. The vectors are perpendicular (orthogonal).
• Negative dot product (A · B < 0): the angle θ is obtuse (between 90° and 180°). The vectors point mostly in opposite directions.

The dot product also captures how large the component of one vector is along the direction of the other. If you normalize one vector (make its length 1), the dot product with the other vector directly gives this signed component.

4. Projections: how much of one vector lies along another?

A common use of the dot product is to project one vector onto another, for example to find how much of a force acts along a specific axis or surface.

The scalar projection of A onto B (also called the component of A along B) is

comp_B(A) = (A · B) / |B|

The vector projection of A onto B is the vector that lies in the direction of B with this length:

proj_B(A) = (A · B / |B|²) B

This calculator focuses on the dot product, magnitudes, and angle, but the same intermediate values can be used to compute projections if needed.

5. Worked example

Suppose you want to find the dot product and angle between A = (2, −1, 3) and B = (4, 0, −2).

5.1 Entering the vectors

• In the calculator, enter A_x = 2, A_y = −1, A_z = 3.
• Enter B_x = 4, B_y = 0, B_z = −2.
• Click “Compute Dot Product”.

5.2 Dot product

A · B = (2)(4) + (−1)(0) + (3)(−2)

= 8 + 0 − 6 = 2.

The dot product is positive but relatively small, so the vectors are slightly more aligned than perpendicular.

5.3 Magnitudes

|A| = √(2² + (−1)² + 3²) = √(4 + 1 + 9) = √14.

|B| = √(4² + 0² + (−2)²) = √(16 + 0 + 4) = √20.

5.4 Angle between A and B

Use the angle formula:

cos θ = (A · B) / (|A||B|) = 2 / (√14 ⋅ √20).

First compute the denominator: √14 ⋅ √20 = √(14 ⋅ 20) = √280.

So cos θ = 2 / √280 ≈ 2 / 16.733 ≈ 0.1195.

Then θ = arccos(0.1195) ≈ 83.1°.

When you enter these values into the calculator, it should return a dot product of 2 and an angle close to 83°, matching the hand calculation (small differences may arise from rounding).

5.5 Optional: projection

If you also want the projection of A onto B, use

proj_B(A) = (A · B / |B|²) B = (2 / 20) B = 0.1 B.

So proj_B(A) = 0.1 (4, 0, −2) = (0.4, 0, −0.2).

6. Comparison: dot product vs related vector operations

The dot product is one of several basic operations on vectors. The table below compares it with a few closely related concepts.

7. Applications of the dot product

7.1 In physics: work and forces

In mechanics, the work done by a constant force F acting over a displacement s is defined as W = F · s. Only the component of the force in the direction of motion contributes to work. If the force is perpendicular to the motion, the dot product is zero and no work is done.

7.2 In computer graphics: lighting and shading

In 3D rendering, the brightness of a surface under directional light often depends on the dot product between a unit surface normal vector and a unit light direction vector. A larger dot product means the surface faces the light more directly and appears brighter, while a negative dot product means the surface faces away from the light.

7.3 In data science: cosine similarity

High-dimensional feature vectors (for example, document embeddings) are often compared using cosine similarity, defined as (A · B) / (|A||B|). This is exactly the same expression used to compute cos θ for the angle between vectors, so cosine similarity is the cosine of the angle between two data vectors.

8. How to use this calculator step by step

1. Identify the components of your two 3D vectors: A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z).
2. Enter A_x, A_y, and A_z into the Vector A input fields.
3. Enter B_x, B_y, and B_z into the Vector B input fields.
4. Click the button to compute the dot product.
5. Read off the dot product, magnitudes, and angle in degrees from the results panel as provided by the page.

9. Limitations and assumptions

• 3D real-valued vectors: The inputs represent 3D vectors with real-number components. Complex vectors and higher dimensions are not handled.
• Zero vectors: If either vector has zero magnitude, the angle is undefined because a zero vector has no direction. The dot product will still compute as 0, but the calculator should indicate that the angle cannot be determined.
• Floating-point precision: Calculations are performed using standard floating-point arithmetic. Very large or very small values may lead to rounding errors; results that should be exactly perpendicular may appear with tiny nonzero dot products due to numerical noise.
• Units: The calculator assumes consistent units for both vectors (for example, all components in meters). If you mix units, the numerical results may not have a clear physical meaning.
• 2D vectors: You can use 2D vectors by setting the third component (z) to 0, but the interface and explanation are written for 3D by default.