Find Vector Perpendicular to Two Vectors Calculator
Vector Cross Product Calculator
Enter the components of two vectors A and B to find a vector perpendicular to both (their cross product A x B).
Result (A x B)
Intermediate Calculations:
a2*b3 = 0
a3*b2 = 0
a3*b1 = 0
a1*b3 = 0
a1*b2 = 0
a2*b1 = 0
The vector perpendicular to A = (a1, a2, a3) and B = (b1, b2, b3) is their cross product A x B = (a2*b3 – a3*b2, a3*b1 – a1*b3, a1*b2 – a2*b1).
| Vector | Component 1 (x) | Component 2 (y) | Component 3 (z) |
|---|---|---|---|
| Vector A | 1 | 2 | 3 |
| Vector B | 4 | 5 | 6 |
| A x B (Perpendicular) | 0 | 0 | 0 |
Table showing input vectors and the resulting perpendicular vector.
Bar chart showing the components of the resultant perpendicular vector A x B.
What is a Find Vector Perpendicular to Two Vectors Calculator?
A "find vector perpendicular to two vectors calculator" is a tool that computes the cross product of two vectors in three-dimensional space. The result of the cross product is a third vector that is orthogonal (perpendicular) to both of the original vectors. This calculator is essential in fields like physics, engineering, computer graphics, and mathematics where directional relationships between vectors are crucial.
Anyone working with 3D geometry or vector quantities can benefit from using this calculator. This includes students learning vector algebra, engineers designing physical systems, physicists analyzing forces and fields, and game developers creating 3D environments. The find vector perpendicular to two vectors calculator simplifies a computation that, while straightforward, can be prone to manual errors.
A common misconception is that there is only one vector perpendicular to two given vectors. While the direction is unique (or its exact opposite), the magnitude of the perpendicular vector depends on the magnitudes of the original vectors and the angle between them. The cross product gives a specific perpendicular vector whose magnitude is also related to the area of the parallelogram formed by the two original vectors.
Find Vector Perpendicular to Two Vectors Calculator Formula and Mathematical Explanation
To find a vector perpendicular to two given vectors A = (a1, a2, a3) and B = (b1, b2, b3), we calculate their cross product, denoted as A x B.
The formula for the cross product is:
A x B = (a2*b3 – a3*b2, a3*b1 – a1*b3, a1*b2 – a2*b1)
This can also be represented as the determinant of a matrix:
A x B =
| i j k |
| a1 a2 a3 |
| b1 b2 b3 |
Where i, j, and k are the standard unit vectors in the x, y, and z directions, respectively.
Expanding the determinant gives:
A x B = i(a2*b3 – a3*b2) – j(a1*b3 – a3*b1) + k(a1*b2 – a2*b1)
A x B = (a2*b3 – a3*b2)i + (a3*b1 – a1*b3)j + (a1*b2 – a2*b1)k
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| A | First input vector | Component units (e.g., m, N) | Real numbers |
| B | Second input vector | Component units (e.g., m, N) | Real numbers |
| a1, a2, a3 | Components of vector A | Component units | Real numbers |
| b1, b2, b3 | Components of vector B | Component units | Real numbers |
| A x B | Resulting perpendicular vector | (Component units)^2 | Real numbers |
Variables involved in the cross product calculation.
Practical Examples (Real-World Use Cases)
Example 1: Finding the Normal to a Plane
Suppose we have two vectors lying on a plane: Vector A = (2, 1, 0) and Vector B = (1, 3, 0). To find a vector normal (perpendicular) to this plane, we use the find vector perpendicular to two vectors calculator (i.e., calculate their cross product).
A = (2, 1, 0), B = (1, 3, 0)
A x B = (1*0 – 0*3, 0*1 – 2*0, 2*3 – 1*1) = (0, 0, 6-1) = (0, 0, 5)
The vector (0, 0, 5) is perpendicular to both A and B, and thus normal to the plane defined by them (which is the xy-plane in this case, so the normal is along the z-axis).
Example 2: Calculating Torque
In physics, torque (τ) is the rotational equivalent of linear force and is defined as the cross product of the position vector (r) from the axis of rotation to the point where the force is applied, and the force vector (F): τ = r x F.
Let r = (1, 2, 0) meters and F = (3, 0, 0) Newtons.
τ = r x F = (2*0 – 0*0, 0*3 – 1*0, 1*0 – 2*3) = (0, 0, -6) Newton-meters.
The torque vector (0, 0, -6) N·m is perpendicular to both the position and force vectors, indicating the axis of rotation and direction of the torque.
How to Use This Find Vector Perpendicular to Two Vectors Calculator
Using the find vector perpendicular to two vectors calculator is simple:
- Enter Vector A Components: Input the values for a1, a2, and a3, which represent the x, y, and z components of the first vector.
- Enter Vector B Components: Input the values for b1, b2, and b3, which represent the x, y, and z components of the second vector.
- Calculate: The calculator will automatically update the results as you type. You can also click the "Calculate" button.
- Read Results: The primary result is the perpendicular vector (A x B), displayed as (x, y, z). Intermediate products are also shown.
- View Table and Chart: The table summarizes the input vectors and the result. The chart visually represents the components of the resultant vector.
- Reset: Click "Reset" to clear the fields to default values.
- Copy Results: Click "Copy Results" to copy the main result, intermediate values, and input vectors to your clipboard.
The find vector perpendicular to two vectors calculator gives you the components of the vector that is orthogonal to both input vectors according to the right-hand rule.
Key Factors That Affect Find Vector Perpendicular to Two Vectors Calculator Results
The output of the find vector perpendicular to two vectors calculator (the cross product) is directly influenced by:
- Components of Vector A (a1, a2, a3): The individual components of the first vector are fundamental to the calculation. Changing any component will alter the resulting perpendicular vector.
- Components of Vector B (b1, b2, b3): Similarly, the components of the second vector are crucial.
- Order of Vectors: The cross product is anti-commutative, meaning A x B = -(B x A). Swapping the order of the vectors will result in a perpendicular vector pointing in the opposite direction but with the same magnitude. Our find vector perpendicular to two vectors calculator computes A x B.
- Magnitude of Input Vectors: The magnitude of the resulting vector |A x B| = |A| |B| sin(θ), where θ is the angle between A and B. Larger magnitudes of A or B generally lead to a larger magnitude of A x B, unless sin(θ) is small.
- Angle Between Vectors (θ): The magnitude of the cross product is maximum when the vectors are perpendicular (θ = 90°, sin(θ) = 1) and zero when they are parallel or anti-parallel (θ = 0° or 180°, sin(θ) = 0).
- Collinearity: If the two vectors are collinear (parallel or anti-parallel), their cross product is the zero vector (0, 0, 0), as there isn't a unique direction perpendicular to both in the sense of the cross product's definition linked to the area of the parallelogram they form. Our find vector perpendicular to two vectors calculator handles this.
Frequently Asked Questions (FAQ)
A: The cross product is a binary operation on two vectors in three-dimensional space. The result is a vector that is perpendicular to both of the original vectors, and its magnitude is equal to the area of the parallelogram spanned by them. Use our find vector perpendicular to two vectors calculator to find it.
A: It means the vector forms a 90-degree angle with each of the two original vectors. The dot product between the perpendicular vector and each of the original vectors is zero.
A: No, the cross product is anti-commutative: A x B = -(B x A). They have the same magnitude but opposite directions. Our find vector perpendicular to two vectors calculator calculates A x B.
A: If the two vectors are parallel (or anti-parallel), their cross product is the zero vector (0, 0, 0). This is because the angle between them is 0 or 180 degrees, and sin(0) = sin(180) = 0.
A: If the components of the original vectors have certain units, the components of the cross product vector will have units that are the product of the units of the original vectors (e.g., if vectors are in meters, the cross product components are in meters-squared, though it often represents a different physical quantity like torque N·m from N and m).
A: The direction is given by the right-hand rule. If you curl the fingers of your right hand from A towards B, your thumb points in the direction of A x B.
A: To use the cross product concept, you typically embed 2D vectors in 3D space by setting their z-components to zero (e.g., (x, y) becomes (x, y, 0)). The cross product will then be a vector along the z-axis.
A: It's used in physics (torque, angular momentum, magnetic force), computer graphics (surface normals), engineering, and geometry to find normals to planes and areas of parallelograms.
Related Tools and Internal Resources
Explore more vector and mathematical calculators:
- Cross Product Explained: A detailed guide on the cross product.
- Dot Product Calculator: Calculate the dot product of two vectors.
- Vector Magnitude Calculator: Find the length of a vector.
- Vector Addition Calculator: Add two or more vectors.
- Plane Equation from Vectors Calculator: Find the equation of a plane given vectors.
- Physics Calculators: A collection of calculators for various physics problems, including those using the find vector perpendicular to two vectors calculator logic.