Find U X V Calculator

Easily calculate the cross product of two 3D vectors u and v using our online find u x v calculator. Get the resultant vector and intermediate steps.

Vector Cross Product Calculator

Enter the components of vectors u and v:

Vector	x-component	y-component	z-component
u	1	2	3
v	4	5	6
u x v	-3	6	-3

Table showing components of vectors u, v, and u x v.

Understanding the Find u x v Calculator and Vector Cross Product

What is the Vector Cross Product (u x v)?

The vector cross product, denoted as u x v, is a binary operation on two vectors u and v in three-dimensional space. The result of the cross product is another vector that is perpendicular to both u and v, and thus normal to the plane containing them. The magnitude of the resulting vector is equal to the area of the parallelogram that the vectors u and v span, and its direction is given by the right-hand rule. Our find u x v calculator helps you compute this resultant vector efficiently.

This operation is widely used in physics (e.g., calculating torque, angular momentum, or the force on a moving charge in a magnetic field), engineering, and computer graphics. Unlike the dot product, which results in a scalar, the cross product results in a vector. The find u x v calculator is essential for students and professionals dealing with 3D vector algebra.

Common misconceptions include confusing the cross product with the dot product or assuming it's commutative (u x v = v x u), which is incorrect; the cross product is anti-commutative (u x v = – v x u).

Find u x v Calculator Formula and Mathematical Explanation

If u = (u_x, u_y, u_z) and v = (v_x, v_y, v_z), the cross product u x v is calculated as:

u x v = (u_yv_z – u_zv_y)i + (u_zv_x – u_xv_z)j + (u_xv_y – u_yv_x)k

Where i, j, and k are the standard unit vectors along the x, y, and z axes, respectively. The components of the resultant vector w = u x v = (w_x, w_y, w_z) are:

• w_x = u_yv_z – u_zv_y
• w_y = u_zv_x – u_xv_z
• w_z = u_xv_y – u_yv_x

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

u x v = | i j k |
| u_x u_y u_z |
| v_x v_y v_z |

The find u x v calculator implements these formulas.

Variables Table

Variable	Meaning	Unit	Typical Range
ux, uy, uz	Components of vector u	Depends on context (e.g., m, m/s)	Real numbers
vx, vy, vz	Components of vector v	Depends on context (e.g., m, m/s)	Real numbers
wx, wy, wz	Components of the resultant vector u x v	Depends on context	Real numbers

Practical Examples (Real-World Use Cases)

Let's see how the find u x v calculator works with examples.

Example 1: Finding a Normal Vector

Suppose we have two vectors in a plane: u = (2, 1, -1) and v = (1, 3, 2). We want to find a vector normal to the plane containing u and v.

Using the find u x v calculator or the formula:

• w_x = (1)(2) – (-1)(3) = 2 – (-3) = 5
• w_y = (-1)(1) – (2)(2) = -1 – 4 = -5
• w_z = (2)(3) – (1)(1) = 6 – 1 = 5

So, u x v = (5, -5, 5). This vector is perpendicular to both u and v.

Example 2: Calculating Torque

Torque (τ) can be calculated 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 x F. Let r = (0.5, 0, 0) meters and F = (0, 10, 0) Newtons.

Using the find u x v calculator logic with u=r and v=F:

• τ_x = (0)(0) – (0)(10) = 0
• τ_y = (0)(0) – (0.5)(0) = 0
• τ_z = (0.5)(10) – (0)(0) = 5

The torque vector is τ = (0, 0, 5) Newton-meters, meaning a torque of 5 Nm around the z-axis. Explore more physics calculators for related tools.

How to Use This Find u x v Calculator

1. Enter Vector u Components: Input the x, y, and z components (u_x, u_y, u_z) of the first vector into the respective fields.
2. Enter Vector v Components: Input the x, y, and z components (v_x, v_y, v_z) of the second vector.
3. View Results: The calculator automatically updates the resultant vector u x v, displaying its components (w_x, w_y, w_z) in the "Result (u x v)" section, along with intermediate calculations.
4. Examine Table and Chart: The table summarizes the components of u, v, and u x v. The chart visualizes the magnitudes of these components.
5. Reset: Click the "Reset" button to clear the inputs and results to default values.
6. Copy Results: Click "Copy Results" to copy the main result, intermediate steps, and input values to your clipboard.

The find u x v calculator provides immediate feedback, making it easy to see how changes in the input vectors affect the cross product. The resulting vector's direction is determined by the right-hand rule relative to u and v.

Key Factors That Affect Find u x v Calculator Results

1. Components of Vector u: The values of u_x, u_y, and u_z directly influence the magnitude and direction of the cross product.
2. Components of Vector v: Similarly, the values of v_x, v_y, and v_z are crucial.
3. Order of Vectors: The cross product is anti-commutative (u x v = – v x u). Swapping the vectors negates the resultant vector. Our find u x v calculator respects this order.
4. Angle Between Vectors: The magnitude of u x v is |u||v|sin(θ), where θ is the angle between u and v. If the vectors are parallel (θ=0° or 180°), the cross product is the zero vector. If they are perpendicular (θ=90°), the magnitude is maximized.
5. Dimensionality: The cross product as defined here is specific to 3-dimensional vectors.
6. Right-Hand Rule: The direction of u x v follows the right-hand rule. If you curl the fingers of your right hand from u to v, your thumb points in the direction of u x v. You might also find our vector magnitude calculator useful.

Frequently Asked Questions (FAQ)

1. What is the find u x v calculator used for?

It's used to calculate the vector cross product of two 3D vectors, u and v, resulting in a vector perpendicular to both, with applications in physics, engineering, and graphics.

2. Is u x v the same as v x u?

No, the cross product is anti-commutative: u x v = – (v x u). The magnitude is the same, but the direction is opposite.

3. What if u and v are parallel?

If u and v are parallel or anti-parallel, their cross product u x v is the zero vector (0, 0, 0) because the angle between them is 0 or 180 degrees, and sin(0) = sin(180) = 0.

4. What does the magnitude of u x v represent?

The magnitude of u x v (|u x v|) represents the area of the parallelogram formed by vectors u and v as adjacent sides.

5. Can I use the find u x v calculator for 2D vectors?

The cross product is inherently a 3D operation. For 2D vectors u=(u_x, u_y) and v=(v_x, v_y), you can embed them in 3D as (u_x, u_y, 0) and (v_x, v_y, 0). The cross product will be (0, 0, u_xv_y – u_yv_x), a vector along the z-axis whose magnitude is related to the area in 2D. You might also be interested in a vector addition calculator.

6. How is the direction of u x v determined?

By the right-hand rule. Point your index finger along u, your middle finger along v, and your thumb will point in the direction of u x v.

7. What are the units of u x v?

The units of u x v are the product of the units of u and v. For example, if u is in meters and v is in Newtons, u x v is in Newton-meters.

8. Does the find u x v calculator handle non-numeric input?

The calculator expects numeric values for the vector components. It includes basic validation to check if the inputs are numbers.