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Orthogonal Vectors Checker

Enter the components of both vectors to test orthogonality.

How it is calculated

For vectors a = (ax, ay, az) and b = (bx, by, bz) the dot product is a·b = axbx + ayby + azbz. The vectors are orthogonal when it equals zero. All six components are required; for 2D vectors, set both z values to 0.

The dot product is displayed with two decimals, but the verdict uses a strict tolerance of 0.000000001, so a dot product of 0.004 shows as 0.00 and still reads Not orthogonal. A zero vector has a dot product of 0 with anything, so the result then reads “Orthogonal (zero vector: dot product is 0 by definition)”; the theorem is stated for nonzero vectors, so treat that as a flag, not as a perpendicular pair. The tool does not check that the vectors have the same units or normalise them. Orthogonality does not depend on vector length.

Worked example

Take a = (1, 2, 3) and b = (4, −5, 2). The products are 1 × 4 = 4, 2 × (−5) = −10 and 3 × 2 = 6. Their sum is 4 − 10 + 6 = 0, so the dot product is 0.00 and the result is Orthogonal. Changing b to (4, 5, 6) gives 4 + 10 + 18 = 32.00, which is not orthogonal.

Try this example →

Common mistakes

  • Typing a 2D vector without the third pair: all six fields are needed, so put 0 in each z component.
  • Confusing the dot product with the cross product: the dot product is a single number and is zero for perpendicular vectors, while the cross product is a vector and is zero for parallel ones.
  • Rounding the inputs, for example 0.333 for 1/3: the dot product then lands a little away from zero and the verdict is Not orthogonal, even if the exact vectors are perpendicular.
  • Reading 0.00 as proof by itself: the display is rounded to two decimals, so check the verdict line, not just the number.

Sources

About this calculator

Two vectors are orthogonal (perpendicular, meeting at exactly 90°) precisely when their dot product is zero: for 3D vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), that's a·b = a₁b₁ + a₂b₂ + a₃b₃. Any nonzero result tells you the angle between them via cosθ = (a·b)/(|a||b|), but zero is the special case that means the vectors share no component in each other's direction.

Engineering, physics, and computer-graphics work leans on this test constantly — resolving forces into independent perpendicular components in statics, checking that a surface normal is perpendicular to a triangle's edges for correct lighting in 3D rendering, or verifying that basis vectors used in a coordinate system are mutually independent — which is why students and developers alike reach for a quick dot-product check rather than computing angles by hand.

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Comments (1)

  • Editorial team

    The check is strict: the verdict says orthogonal only when the dot product is within one billionth of zero. Vectors from measured or rounded data, such as (1, 0, 0) and (0.001, 1, 0), show 0.00 yet read as not orthogonal, because the true dot product is 0.001. For real data, compute the angle from the dot product and the lengths instead. To test a set of three basis vectors, check all three pairs. Which kind of vectors are you testing?

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