To calculate a dot product, multiply each component of one vector by the component in the same position in the other vector, then add those products. For example, (1, 2, 3) · (4, −1, 2) = 1×4 + 2×(−1) + 3×2 = 8. The result is a single number, called a scalar—not another vector.
How to calculate a dot product from coordinates
For two real vectors with the same number of components, the coordinate formula is:
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u · v = u₁v₁ + u₂v₂ + … + uₙvₙ
Each component is multiplied by the component in the matching position. Add every product, including negative ones. The University of Nebraska–Lincoln demonstrates the method with (−2, 0, 1) · (3, 2, −4) = (−2)(3) + (0)(2) + (1)(−4) = −10 in its dot product lesson.
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Calculate (2, −1) · (3, 4) by pairing entries in order:
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- Multiply the first pair:
2 × 3 = 6. - Multiply the second pair:
(−1) × 4 = −4. - Add the products:
6 + (−4) = 2.
So (2, −1) · (3, 4) = 2. Adding the original components first, or stopping at the list of pairwise products, does not complete the dot product: the definition requires adding the products.
Check the vector dimensions
The coordinate formula pairs entries by position, so both vectors must have the same number of components and belong to the same coordinate space, such as ℝ² or ℝ³. If one vector has two entries and the other has three, this standard coordinate dot product is not defined.
Why the answer is a number
The operation ends by summing the component products, and that sum is one scalar. It does not return the component-wise products as a new vector. For example, the intermediate products for (2, −1) · (3, 4) are 6 and −4; their sum, 2, is the dot product.
What the dot product says about direction
The geometric formula for two vectors is u · v = |u||v| cos θ, where |u| and |v| are their lengths and θ is the angle between them. It gives the same scalar as the coordinate calculation. Use the coordinate formula when the components are given; the magnitude-angle formula is useful when the lengths and angle are known or when interpreting how the vectors point relative to each other. MIT’s Vector Calculus: Dot Product explains the geometric relationship.
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Positive, zero, and negative results
When both vectors are nonzero, the sign reflects the angle between them:
- Positive: the angle is acute, so the vectors point generally in the same direction.
- Zero: the angle is 90°, so the vectors are perpendicular, or orthogonal.
- Negative: the angle is obtuse, so the vectors point generally in opposing directions.
The zero-vector exception matters: a zero vector has dot product zero with every vector, but it has no direction and does not form a usual angle with another vector. Therefore, a zero result establishes perpendicularity only when both vectors are nonzero. See the University of Nebraska–Lincoln’s discussion of orthogonality.
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Finding the angle from a dot product
If both vectors are nonzero, rearrange the geometric formula to find the angle:
θ = arccos((u · v) / (|u||v|))
Use a calculator’s degree mode if you want the answer in degrees, or radian mode if you want radians. The dot product itself is not an angle; the inverse cosine gives the angle after the dot product is divided by the product of the vector lengths.
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How the dot product relates to length
Taking the dot product of a vector with itself gives its squared length: u · u = ||u||². Therefore, ||u|| = √(u · u). A self-dot-product can never be negative; it is zero only when the vector is the zero vector. This is a useful check when working through a calculation. The University of Nebraska–Lincoln also describes the dot product’s symmetry and linearity properties in its lesson: u · v = v · u, sums distribute, and scalar factors can be pulled out.
What a dot product is useful for
The magnitude-angle form measures directional alignment scaled by the vectors’ lengths. A related projection interpretation is especially clear for a unit vector u: a · u = ||a|| cos θ gives the signed component of a in the direction of u. The University of Minnesota’s Math Insight explanation develops this interpretation.
In physics, the dot product connects force and displacement to work: the component of force in the direction of motion contributes to the work. Lamar University’s Calculus II dot product notes describe this application.
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A free scan shows the junk files, broken settings and background clutter dragging Windows down - then fixes them in one click.Free scan · Windows 10 & 11The dot product also does not depend on which coordinate axes are used to describe the vectors. MIT OpenCourseWare notes that “the dot product is invariant under rotation of coordinates” in its section on the dot product.
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