3VDP

Math & Statistics

3D Vector Dot Product Calculator

Calculate the dot product of two 3D vectors, each component contribution, both magnitudes, cosine of the included angle, scalar projection, squared endpoint distance, and Euclidean distance.

Vector A xVector A yVector A zVector B xVector B yVector B z

A dot B-

Magnitude of A-

Magnitude of B-

Cosine of included angle-

Scalar projection of A onto B-

Squared Euclidean distance-

Euclidean distance between endpoints-

Absolute dot-product reference-

Decision view

Signed component contributions and cosine alignment gauge

Signed component contributions and cosine alignment gaugeDiverging x, y, and z products reconcile to the dot product while a normalized gauge separates direction from magnitude.

Exact scenario comparisonVector B z changes while all other entered assumptions remain constant.

Vector B z	A dot B	Magnitude of A	Magnitude of B	Cosine of included angle	Scalar projection of A onto B	Squared Euclidean distance	Euclidean distance between endpoints	Absolute dot-product reference

How to use 3D Vector Dot Product Calculator

Enter all three components for vectors A and B in one coordinate basis.
Inspect the three signed component products.
Use magnitudes and cosine to separate size from direction.
Read scalar projection and endpoint distance as distinct geometric outputs.

Calculator guide

Understanding 3D Vector Dot Product Calculator

The dot product combines corresponding vector components into one signed alignment measure. This calculator also exposes vector magnitudes, cosine similarity, scalar projection, and endpoint distance so alignment is not confused with size.

Components contribute Each axis can align or oppose.

Magnitude affects dot product Large vectors can produce a large total.

Cosine isolates direction Normalization removes vector size.

Distance differs Endpoint separation is not the dot product.

Calculation method

How the calculation works

Multiply corresponding 3D components and sum them, then reconcile magnitudes, cosine similarity, scalar projection, and endpoint distance. In the 3D Vector Dot Product Calculator, the live scenario varies vector b z and tracks a dot b while the remaining results preserve the reconciliation path. Multiply matching x, y, and z components and sum them. Divide the dot product by the product of magnitudes for cosine similarity, or by the magnitude of vector B for the scalar projection of A onto B.

Detailed calculation process

Add component products and separate alignment from magnitude

The defaults compare A = (3, 4, 2) with B = (5, -1, 6).

General formula: A dot B = AxBx + AyBy + AzBz; |A| = sqrt(Ax^2+Ay^2+Az^2); cos(theta) = (A dot B)/(|A||B|); comp_B(A) = (A dot B)/|B| Positive component products align, negative products oppose, and their sum forms the dot product. Normalizing by both magnitudes isolates angular alignment.

What each symbol means

Ax, Ay, Az Components of vector A in compatible coordinate units.

Bx, By, Bz Components of vector B in the same coordinate basis.

A dot B Signed dot product in squared component units.

|A|, |B| Vector magnitudes in component units.

theta Included angle between nonzero vectors, measured in angular units.

comp_B(A) Signed scalar projection of A onto B in component units.

Worked substitution with the default inputs

1. Multiply corresponding components: x: 3x5 = 15; y: 4x(-1) = -4; z: 2x6 = 12 The y components oppose while x and z components align.

2. Add the dot product: A dot B = 15 - 4 + 12 = 23 The positive total indicates an acute included angle for nonzero vectors.

3. Calculate both magnitudes: |A| = sqrt(9+16+4) = sqrt(29) = 5.385165; |B| = sqrt(25+1+36) = sqrt(62) = 7.874008 Magnitude measures size independently of direction.

4. Normalize the alignment: cos(theta) = 23/(sqrt(29)sqrt(62)) = 0.542417; comp_B(A) = 23/sqrt(62) = 2.921003 Cosine is unitless while scalar projection returns to component units.

5. Reconcile endpoint distance: |A-B| = sqrt[(3-5)^2+(4+1)^2+(2-6)^2] = sqrt(45) = 6.708204 Endpoint distance answers a different geometric question from angular alignment.

The defaults give A dot B = 23, magnitudes 5.385 and 7.874, cosine similarity 0.5424, scalar projection 2.921, and endpoint distance 6.708.

Alignment anatomy

See which axis drives the dot product

A diverging contribution view makes cancellation and reinforcement explicit.

Positive bars Axes pointing in the same signed direction.

Negative bars Axes opposing one another.

Total The sum A dot B.

Cosine gauge Normalized directional alignment from -1 to 1.

Worked situations

Practical examples

The x contribution is +15 and the y contribution is -4.
A positive dot product of 23 implies an acute angle for these nonzero vectors.
The unitless cosine 0.5424 shows moderate directional alignment.

Better inputs

Useful tips

Use the same axis orientation and component units for both vectors.
Inspect signed contributions before interpreting the total.
Treat near-zero magnitudes as a division warning.

Before relying on the result

Limitations and common mistakes

Cosine and scalar projection are undefined for zero-length vectors.
Component units and coordinate systems must be compatible.
The calculator does not perform cross products, rotations, uncertainty propagation, or basis transformations.

Reference

Key terms

Dot product: Sum of corresponding component products.
Cosine similarity: Dot product normalized by both vector magnitudes.
Scalar projection: Signed length of one vector along another vector's direction.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

What does a negative dot product mean?

For nonzero vectors, their included angle is obtuse and opposing component effects dominate.

What does a zero dot product mean?

Nonzero vectors are orthogonal, though zero-vector cases need separate handling.

Why can cosine differ from the raw dot product?

Cosine removes magnitude by dividing by |A||B|.

Is endpoint distance an angle measure?

No. It measures the Euclidean separation between the two vector endpoints.