Math
Dot Product Calculator
Enter two vectors and mathcheck multiplies matching components to get the dot (scalar) product.
It then uses the result to find the angle between the vectors, tells you whether they are perpendicular, parallel, acute, or obtuse, and gives the scalar projection of the first vector onto the second.
Horizontal component of Vector a.
Vertical component of Vector a.
Horizontal component of Vector b.
Vertical component of Vector b.
Try an example
Result
Dot product a · b
—
- Angle between a and b
- —
- Relationship
- —
- |a|
- —
- |b|
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- Scalar projection
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the dot product is
The Multiply two vectors piece by piece and add the results to get one number. (also called the scalar or inner product) combines two A quantity with both a size and a direction, often drawn as an arrow. into a single number. Algebraically it is the sum of the products of matching components. Geometrically it equals |a| |b| cos θ, so it measures how much the two vectors point in the same direction: positive for an acute angle, zero for perpendicular vectors, negative for an obtuse angle.
⟨3, 4⟩ · ⟨2, -1⟩
3(2) + 4(-1) = 6 - 4 = 2. |a| = 5, |b| = √5. cos θ = 2 / (5√5) ≈ 0.1789, so θ ≈ 79.70°. The dot product is positive, so the angle is acute.
Reading the sign
| a · b | Angle | Meaning |
|---|---|---|
| positive | 0° ≤ θ < 90° | vectors lean the same way |
| zero | θ = 90° | vectors are orthogonal (perpendicular) |
| negative | 90° < θ ≤ 180° | vectors lean opposite ways |
| |a| |b| | θ = 0° | parallel, same direction |
| -|a| |b| | θ = 180° | parallel, opposite direction |
How to use this calculator
- Choose 2D or 3D.
- Enter the components of vector a and vector b.
- Read a · b, then the angle between the vectors and the relationship badge.
- Use the scalar projection when you need the component of a in the direction of b.
- Open Show the work to see each product and the arccos step.
How to read the answer
A dot product of 0 means the vectors are perpendicular, the most common thing homework asks you to check. A positive value means the angle between them is less than 90°, negative means more than 90°. The size of the number depends on both lengths, so compare the cosine (dot product divided by the The length or size of a vector, ignoring which way it points.) if you want a length-free measure of alignment.
Common mistakes and edge cases
- Multiplying across instead of matching: a1·b2 is part of the cross product, not the dot product.
- Expecting a vector answer. The dot product is a single number.
- Forgetting the z-components in 3D.
- Using degrees and An angle unit where a full turn is 2π instead of 360 degrees. inconsistently when checking |a| |b| cos θ by hand.
- The angle is undefined when either vector is the zero vector, even though the dot product is 0.
Worked examples
Acute angle
⟨3, 4⟩ · ⟨2, -1⟩
Dot product a · b
2
Orthogonal vectors
⟨1, 2⟩ · ⟨-2, 1⟩
Dot product a · b
0
3D dot product
⟨1, 2, 3⟩ · ⟨4, -5, 6⟩
Dot product a · b
12
Opposite directions
⟨2, 2⟩ · ⟨-1, -1⟩
Dot product a · b
-4
Decimal components
⟨1.5, 0.5⟩ · ⟨2, 4⟩
Dot product a · b
5
Zero vector
⟨0, 0⟩ · ⟨3, 1⟩
Dot product a · b
0
Frequently asked questions
How do you calculate the dot product?+
Multiply matching components and add: ⟨a1, a2⟩ · ⟨b1, b2⟩ = a1·b1 + a2·b2. In 3D add a3·b3 as well.
What does a dot product of zero mean?+
The vectors are orthogonal (perpendicular), as long as neither is the zero vector. This is the standard test for perpendicular vectors.
How do I find the angle between two vectors with the dot product?+
Use cos θ = (a · b) / (|a| |b|), then θ = arccos of that value. The calculator shows both the cosine and the angle.
Is the dot product a vector or a scalar?+
A scalar (a single number). The cross product is the one that returns a vector.
Can the dot product be negative?+
Yes. A negative dot product means the angle between the vectors is greater than 90°, so they point in generally opposite directions.
What is the dot product used for?+
Finding angles between vectors, testing perpendicularity, computing work (force · displacement) in physics, and projecting one vector onto another.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026