Orthogonal Vectors Calculator

Enter the vector components to determine if they are orthogonal (perpendicular) or not by calculating their dot product.

Vector a
⟨ , ⟩
Vector b
⟨ , ⟩

Quick Examples

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Solved Problems

The following are examples of problems solved by the calculator.

Determine whether the vectors ⟨2, 3⟩ and ⟨-3, 2⟩ are orthogonal or not.

Result

The given vectors ARE orthogonal.

To confirm this, we calculate their dot product:

$$ \displaystyle \vec{a} \cdot \vec{b} = (2)(-3) + (3)(2) = 0 $$

Since the dot product is equal to zero, the vectors are perpendicular, meaning the angle between them is 90°.

Cartesian plane graph of two mutually orthogonal vectors.
Graph of the vectors
Find out if the vectors \(\langle \sqrt{3}, 1\rangle\) and \(\langle 1, \sqrt{3}\rangle\) are perpendicular to each other.

Result

The given vectors are NOT orthogonal.

To confirm this, we calculate their dot product:

$$ \displaystyle \vec{a} \cdot \vec{b} = \left(\sqrt{3}\right)(1) + (1)\left(\sqrt{3}\right) = 2 \sqrt{3} \neq 0 $$

Since the dot product is non-zero, the vectors are not perpendicular (the angle between them is not 90°).

In particular, the angle between them is:

$$ \displaystyle \theta = 30^\circ $$
Cartesian plane graph of two non-orthogonal vectors.
Graph of the vectors
Determine whether the three-dimensional vectors ⟨2, -1, 3⟩ and ⟨1, 5, 1⟩ are orthogonal or not.

Result

The given vectors ARE orthogonal.

To confirm this, we calculate their dot product:

$$ \displaystyle \vec{a} \cdot \vec{b} = (2)(1) + (-1)(5) + (3)(1) = 0 $$

Since the dot product is equal to zero, the vectors are perpendicular, meaning the angle between them is 90°.

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Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

Last updated: (5 days ago)