Find The Vector Component Of U Orthogonal To V Calculator

Enter the components of vectors u and v to find the component of u orthogonal to v.

What is the Vector Component of u Orthogonal to v?

The vector component of u orthogonal to v, often denoted as w or u_⊥v, is the part of vector u that is perpendicular to vector v. It is found by subtracting the projection of u onto v (the component of u parallel to v) from u itself. Imagine vector u as the sum of two vectors: one parallel to v and one orthogonal to v. This calculator finds the orthogonal part.

This concept is fundamental in linear algebra, physics (like finding forces perpendicular to a surface), and computer graphics. Anyone studying or working with vectors and their interactions will find the vector component of u orthogonal to v calculator useful.

A common misconception is that the orthogonal component is simply u minus v. In reality, it's u minus the projection of u onto v, which scales v based on how much u aligns with v.

Vector Component of u Orthogonal to v Formula and Mathematical Explanation

The formula to find the component of vector u orthogonal to vector v is:

w = u – proj_v(u)

Where:

• w is the component of u orthogonal to v.
• u is the original vector.
• proj_v(u) is the projection of vector u onto vector v.

The projection of u onto v is calculated as:

proj_v(u) = ((u • v) / ||v||²) * v

Here:

• u • v is the dot product of u and v (u_xv_x + u_yv_y + u_zv_z).
• ||v||² is the squared magnitude of v (v_x² + v_y² + v_z²).
• v is the vector v itself.

So, the full formula for the components of w = (w_x, w_y, w_z) is:

w_x = u_x – [((u • v) / ||v||²) * v_x]

w_y = u_y – [((u • v) / ||v||²) * v_y]

w_z = u_z – [((u • v) / ||v||²) * v_z]

This is valid as long as v is not the zero vector (||v||² ≠ 0).

Variables Table

Variable	Meaning	Unit	Typical range
u = (ux, uy, uz)	Vector u	Dimensionless (or units of quantity)	Real numbers
v = (vx, vy, vz)	Vector v	Dimensionless (or units of quantity)	Real numbers (not zero vector)
u • v	Dot product of u and v	(Units of u) * (Units of v)	Real number
||v||^2	Squared magnitude of v	(Units of v)^2	Positive real number
projv(u)	Projection of u onto v	Same units as u	Vector
w	Component of u orthogonal to v	Same units as u	Vector

Variables used in the vector component of u orthogonal to v calculator.

Practical Examples (Real-World Use Cases)

Example 1: Force Decomposition

Imagine a force vector F = (3, 5, 2) N acting on an object, and we want to find the component of this force perpendicular to a ramp represented by the direction vector d = (4, 1, 0).

• u = (3, 5, 2)
• v = (4, 1, 0)
• u • v = (3*4) + (5*1) + (2*0) = 12 + 5 + 0 = 17
• ||v||² = 4² + 1² + 0² = 16 + 1 + 0 = 17
• proj_v(u) = (17 / 17) * (4, 1, 0) = 1 * (4, 1, 0) = (4, 1, 0)
• w = u – proj_v(u) = (3, 5, 2) – (4, 1, 0) = (-1, 4, 2) N

The component of the force F orthogonal to the direction d is (-1, 4, 2) N.

Example 2: Graphics and Projections

In computer graphics, you might want to decompose a vector u = (5, 8, 3) relative to a view direction v = (1, 1, 1).

• u = (5, 8, 3)
• v = (1, 1, 1)
• u • v = (5*1) + (8*1) + (3*1) = 5 + 8 + 3 = 16
• ||v||² = 1² + 1² + 1² = 1 + 1 + 1 = 3
• proj_v(u) = (16 / 3) * (1, 1, 1) = (16/3, 16/3, 16/3) ≈ (5.33, 5.33, 5.33)
• w = u – proj_v(u) = (5, 8, 3) – (16/3, 16/3, 16/3) = (-1/3, 8/3, -7/3) ≈ (-0.33, 2.67, -2.33)

The component of u orthogonal to v is approximately (-0.33, 2.67, -2.33).

How to Use This Vector Component of u Orthogonal to v Calculator

1. Enter Vector u Components: Input the x, y, and z components (u_x, u_y, u_z) of vector u into the first row of input fields. If you are working in 2D, set u_z and v_z to 0.
2. Enter Vector v Components: Input the x, y, and z components (v_x, v_y, v_z) of vector v into the second row of input fields. Ensure v is not the zero vector (0, 0, 0).
3. Calculate: The calculator will automatically update the results as you type. You can also click the "Calculate" button.
4. Read Results: The primary result shows the components of the orthogonal vector w. Intermediate results like the dot product, squared magnitude of v, and the projection vector are also displayed. The chart visualizes the x-y components.
5. Reset: Click "Reset" to return to the default values.
6. Copy: Click "Copy Results" to copy the main result and intermediate values to your clipboard.

The vector component of u orthogonal to v calculator helps visualize how u decomposes relative to v.

Key Factors That Affect the Results

• Components of u: The magnitude and direction of u directly determine the starting vector being decomposed.
• Components of v: The direction of v defines the line onto which u is projected, and thus the direction of the parallel component and, consequently, the orthogonal one. Its magnitude also scales the projection.
• Angle Between u and v: The dot product (u • v = ||u|| ||v|| cos θ) is influenced by the angle θ between the vectors. If u and v are already orthogonal, the projection is zero, and the orthogonal component is u itself. If they are parallel, the orthogonal component is zero.
• Magnitude of v: While the direction of the projection depends only on v's direction, the magnitude of v appears in the denominator ||v||². A larger ||v|| (for the same direction) would make the scalar part of the projection smaller if the dot product remains constant, but the dot product also involves ||v||. The key is that v must not be the zero vector.
• Dimensionality: Whether you are working in 2D or 3D (or higher dimensions) changes the number of components but not the underlying principle. Our vector component of u orthogonal to v calculator handles 3D, and 2D by setting z-components to zero.
• Zero Vector v: If v is the zero vector, ||v||² is zero, and the projection (and thus the orthogonal component) is undefined as division by zero occurs. Our calculator should handle this edge case.

For more on vector operations, check out our dot product calculator or learn about linear algebra basics.

Frequently Asked Questions (FAQ)

Q: What does it mean for a vector to be orthogonal to another?

A: Two vectors are orthogonal (or perpendicular) if their dot product is zero. The component of u orthogonal to v is a vector w such that w • v = 0.

Q: What happens if vector v is the zero vector?

A: If v is the zero vector (0, 0, 0), its squared magnitude ||v||² is 0. Division by zero is undefined, so the projection and the orthogonal component are undefined. Our vector component of u orthogonal to v calculator will show an error or NaN in such cases.

Q: Can I use this calculator for 2D vectors?

A: Yes, simply set the z-components (u_z and v_z) to 0, and the calculation will be correct for 2D vectors.

Q: Is the orthogonal component always shorter than u?

A: Not necessarily. If u is already nearly orthogonal to v, the projection will be small, and the orthogonal component will be close to u in magnitude. If u is nearly parallel to v, the projection will be close to u, and the orthogonal component will be small.

Q: What is the relationship between u, proj_v(u), and w?

A: The original vector u is the sum of its projection onto v and its component orthogonal to v: u = proj_v(u) + w. These two components (proj_v(u) and w) are orthogonal to each other.

Q: How is this different from the cross product?

A: The cross product of two vectors (in 3D) results in a vector orthogonal to BOTH original vectors. The component of u orthogonal to v is a vector in the plane defined by u and v (if they are not collinear) that is orthogonal to v and, when added to proj_v(u), gives u. You might use our cross product calculator for that.

Q: Why is it called the "component" if it's a vector?

A: "Component" here refers to a vector part of u. Vector u is decomposed into two vector components: one parallel to v (the projection) and one orthogonal to v.

Q: Where is the concept of orthogonal components used?

A: It's used in physics (e.g., resolving forces), engineering (e.g., stress analysis), computer graphics (e.g., lighting and camera models), and many areas of mathematics like linear algebra and vector calculus.

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