Find The Orthogonal Vector Calculator

Easily find a vector orthogonal (perpendicular) to two given 3D vectors using our Orthogonal Vector Calculator. We use the cross product method.

Calculate Orthogonal Vector

Input Vectors and Resulting Orthogonal Vector

Vector	x	y	z	Magnitude
A	1	2	3
B	4	5	6
C (A × B)

What is an Orthogonal Vector Calculator?

An Orthogonal Vector Calculator is a tool used to find a vector that is perpendicular (at a 90-degree angle) to two other given vectors in three-dimensional space. The most common way to find such a vector is by calculating the cross product of the two initial vectors. If you have two vectors, A and B, their cross product, A × B, results in a vector C that is orthogonal to both A and B.

This calculator is useful for students, engineers, physicists, and anyone working with vector mathematics, particularly in fields like linear algebra, computer graphics, and physics (e.g., calculating torque or the direction of magnetic force). It simplifies the process of finding a normal vector to a plane defined by two vectors.

Common misconceptions include thinking that there’s only one unique orthogonal vector. While the cross product gives one such vector, any scalar multiple of it (including the opposite direction) is also orthogonal to the original two vectors. Our Orthogonal Vector Calculator provides the result of the standard cross product.

Orthogonal Vector Formula and Mathematical Explanation

Given two vectors in 3D space, A = (x1, y1, z1) and B = (x2, y2, z2), the vector C orthogonal to both A and B is found using the cross product C = A × B.

The components of the resulting vector C = (Cx, Cy, Cz) are calculated as follows:

• C_x = (y1 * z2) – (z1 * y2)
• C_y = (z1 * x2) – (x1 * z2)
• C_z = (x1 * y2) – (y1 * x2)

This can also be represented as the determinant of a matrix:

C = | i j k |

| x1 y1 z1 |

| x2 y2 z2 |

Where i, j, and k are the unit vectors along the x, y, and z axes, respectively.

To verify that C is orthogonal to A and B, their dot products must be zero: A · C = 0 and B · C = 0.

Variables Table

Variable	Meaning	Unit	Typical Range
x1, y1, z1	Components of vector A	Dimensionless or spatial units	Any real number
x2, y2, z2	Components of vector B	Dimensionless or spatial units	Any real number
Cx, Cy, Cz	Components of the orthogonal vector C	Dimensionless or spatial units	Calculated
|A|, |B|, |C|	Magnitudes of vectors A, B, and C	Same as components	Non-negative real numbers

Practical Examples (Real-World Use Cases)

Let’s look at how the Orthogonal Vector Calculator works with examples.

Example 1: Finding a Normal Vector to a Plane

Suppose you have two vectors lying on a plane: Vector A = (2, 3, 1) and Vector B = (1, -1, 4). We want to find a vector normal (orthogonal) to this plane.

Using the Orthogonal Vector Calculator with A=(2, 3, 1) and B=(1, -1, 4):

• Cx = (3 * 4) – (1 * -1) = 12 – (-1) = 13
• Cy = (1 * 1) – (2 * 4) = 1 – 8 = -7
• Cz = (2 * -1) – (3 * 1) = -2 – 3 = -5

The orthogonal vector C is (13, -7, -5). You can verify by checking dot products: A·C = 2*13 + 3*(-7) + 1*(-5) = 26 – 21 – 5 = 0, and B·C = 1*13 + (-1)*(-7) + 4*(-5) = 13 + 7 – 20 = 0.

Example 2: Physics – Torque Calculation

In physics, torque (τ) can be defined as the cross product of the position vector (r) from the axis of rotation to the point where force is applied, and the force vector (F): τ = r × F. The torque vector is orthogonal to both r and F.

If r = (1, 1, 0) meters and F = (0, 10, 0) Newtons:

Using the Orthogonal Vector Calculator principles:

• τx = (1 * 0) – (0 * 10) = 0
• τy = (0 * 0) – (1 * 0) = 0
• τz = (1 * 10) – (1 * 0) = 10

The torque vector τ is (0, 0, 10) Newton-meters, which is along the z-axis, orthogonal to r and F which lie in the xy-plane.

How to Use This Orthogonal Vector Calculator

1. Enter Vector A Components: Input the x, y, and z components (x1, y1, z1) of the first vector into the respective fields.
2. Enter Vector B Components: Input the x, y, and z components (x2, y2, z2) of the second vector.
3. Real-time Calculation: The calculator automatically updates the results as you type. You can also click “Calculate”.
4. View Results: The primary result is the orthogonal vector C = (Cx, Cy, Cz). Intermediate results show the components and magnitudes of A, B, and C, as well as the dot products A·C and B·C (which should be close to zero).
5. Check the Table and Chart: The table summarizes the vectors and their magnitudes, while the chart visually compares the magnitudes.
6. Reset: Click “Reset” to go back to default values.
7. Copy Results: Click “Copy Results” to copy the main orthogonal vector and other details to your clipboard.

The dot products A·C and B·C are displayed to confirm orthogonality; they should be very close to zero (allowing for minor floating-point inaccuracies).

Key Factors That Affect Orthogonal Vector Results

• Components of Input Vectors: The x, y, and z values of vectors A and B directly determine the components, direction, and magnitude of the resulting orthogonal vector C. Small changes in input can lead to significant changes in the output.
• Order of Vectors (A × B vs B × A): The cross product is anti-commutative, meaning A × B = -(B × A). Reversing the order of vectors will result in an orthogonal vector pointing in the opposite direction but with the same magnitude. Our Orthogonal Vector Calculator calculates A × B.
• Collinearity of Vectors: If vectors A and B are collinear (parallel or anti-parallel, meaning one is a scalar multiple of the other), their cross product is the zero vector (0, 0, 0). The Orthogonal Vector Calculator will show this.
• Magnitude of Input Vectors: The magnitude of the orthogonal vector C is |C| = |A| |B| sin(θ), where θ is the angle between A and B. Larger magnitudes of A or B, or an angle closer to 90 degrees, result in a larger magnitude for C.
• Zero Vector Input: If either A or B is the zero vector (0, 0, 0), the cross product (and thus the orthogonal vector C) will also be the zero vector.
• Numerical Precision: When dealing with floating-point numbers, the calculated dot products A·C and B·C might be very small numbers close to zero (e.g., 1e-14) instead of exactly zero due to precision limitations. Our Orthogonal Vector Calculator handles these.

Frequently Asked Questions (FAQ)

What does it mean for vectors to be orthogonal?

Two vectors are orthogonal if they are perpendicular to each other, meaning the angle between them is 90 degrees. Their dot product is zero.

Is the orthogonal vector unique?

The direction of the vector given by A × B is unique (up to scaling), but any scalar multiple of A × B is also orthogonal to A and B. B × A gives a vector in the opposite direction.

What if the two input vectors are parallel?

If vectors A and B are parallel (or anti-parallel), their cross product is the zero vector (0, 0, 0). The Orthogonal Vector Calculator will output (0, 0, 0).

Can I use this calculator for 2D vectors?

This calculator is designed for 3D vectors. For 2D vectors A=(x1, y1) and B=(x2, y2), you can represent them as 3D vectors (x1, y1, 0) and (x2, y2, 0) and use the calculator. The cross product will be along the z-axis.

What is the magnitude of the orthogonal vector?

The magnitude of C = A × B is |C| = |A| |B| sin(θ), where θ is the angle between A and B. This magnitude is equal to the area of the parallelogram formed by A and B.

How do I know if the result from the Orthogonal Vector Calculator is correct?

Check the dot products A·C and B·C displayed by the calculator. They should be zero or very close to zero.

What if I input non-numeric values?

The calculator expects numerical values for the vector components. Non-numeric input will be treated as invalid or zero, depending on the browser, but it’s best to enter numbers.

Can the orthogonal vector be the zero vector?

Yes, if the input vectors are parallel or if one or both are zero vectors, the orthogonal vector (from the cross product) will be the zero vector.