Find The Orthogonal Projection Calculator

Easily calculate the orthogonal projection of one vector onto another and visualize the result.

Calculate Orthogonal Projection

What is Orthogonal Projection?

The orthogonal projection of a vector a onto a non-zero vector b is the shadow or image of vector a cast onto the line containing vector b, as if by a light source perpendicular to vector b. It’s essentially the component of vector a that lies in the direction of vector b.

Imagine dropping a perpendicular line from the head of vector a onto the line defined by vector b. The point where this perpendicular line intersects the line of b, along with the origin, defines the orthogonal projection vector.

Who should use it? Orthogonal projection is a fundamental concept in linear algebra, physics, engineering, computer graphics, and data science. It’s used in:

• Physics: To find the component of a force along a certain direction.
• Computer Graphics: For 3D to 2D projections and shadow calculations.
• Data Science: In techniques like Principal Component Analysis (PCA) and regression.
• Engineering: To resolve forces or velocities into components.

Common misconceptions include confusing orthogonal projection (a vector) with scalar projection (a length), or thinking the projection is always shorter than the original vector (it can be if the angle is large, but its magnitude depends on the cosine of the angle).

Orthogonal Projection Formula and Mathematical Explanation

Given two vectors a = (a_x, a_y) and b = (b_x, b_y) in 2D, the orthogonal projection of a onto b (proj_b a) is found using the following steps:

1. Calculate the dot product of a and b:
   a · b = a_xb_x + a_yb_y
2. Calculate the squared magnitude of b:
   ||b||² = b_x² + b_y²
3. Calculate the scalar multiple:
   This scalar is (a · b) / ||b||². It represents how much vector b needs to be scaled to get the projection.
4. Multiply the scalar by vector b:
   proj_b a = ((a · b) / ||b||²) * b = (((a · b) / ||b||²) * b_x, ((a · b) / ||b||²) * b_y)

The formula for the orthogonal projection is:

proj_b a = ((a · b) / ||b||²) * b

This formula gives us a vector that lies along the direction of b and has a magnitude equal to the scalar projection of a onto b.

Variables Table

Variable	Meaning	Unit	Typical Range
ax, ay	Components of vector a	Dimensionless or spatial units	Any real number
bx, by	Components of vector b	Dimensionless or spatial units	Any real number (b cannot be the zero vector)
a · b	Dot product of a and b	Depends on units of a and b	Any real number
||b||²	Squared magnitude of b	Depends on units of b	Positive real number
projb a	Orthogonal projection of a onto b	Same as a and b	Vector with real components

Practical Examples (Real-World Use Cases)

Example 1: Basic Vectors

Let vector a = (2, 3) and vector b = (4, 1).

1. a · b = (2 * 4) + (3 * 1) = 8 + 3 = 11
2. ||b||² = 4² + 1² = 16 + 1 = 17
3. Scalar = 11 / 17 ≈ 0.647
4. proj_b a = (11/17) * (4, 1) = (44/17, 11/17) ≈ (2.588, 0.647)

The orthogonal projection of (2, 3) onto (4, 1) is approximately (2.588, 0.647).

Example 2: Force Component

A force F = (5, 10) Newtons is acting on an object moving along a direction given by vector d = (3, 4). Find the component of the force along the direction of motion (the orthogonal projection of F onto d).

1. F · d = (5 * 3) + (10 * 4) = 15 + 40 = 55
2. ||d||² = 3² + 4² = 9 + 16 = 25
3. Scalar = 55 / 25 = 2.2
4. proj_d F = 2.2 * (3, 4) = (6.6, 8.8) Newtons

The component of the force F along the direction d is (6.6, 8.8) Newtons. The magnitude of this component is √(6.6² + 8.8²) = √121 = 11 Newtons.

How to Use This Orthogonal Projection Calculator

1. Enter Vector a Components: Input the x (a_x) and y (a_y) components of the vector you want to project.
2. Enter Vector b Components: Input the x (b_x) and y (b_y) components of the vector onto which you are projecting. Vector b cannot be the zero vector (0,0).
3. View Results: The calculator automatically updates the “Projection Vector”, “Dot Product”, “Magnitude of b Squared”, and “Scalar” as you type.
4. See Visualization: The SVG chart updates to show vector a (blue), vector b (red), and the calculated orthogonal projection (green).
5. Reset: Click “Reset” to return to default values.
6. Copy: Click “Copy Results” to copy the main results and intermediate values to your clipboard.

The primary result shows the components of the orthogonal projection vector. The intermediate results help you understand the steps of the calculation.

Key Factors That Affect Orthogonal Projection Results

1. Components of Vector a: Changing the length or direction of vector a directly changes the vector being projected, thus altering the projection.
2. Components of Vector b: The direction of vector b defines the line onto which a is projected. Changing b changes this line and thus the projection. The magnitude of b also affects the intermediate scalar, but the final projection vector’s direction depends only on b‘s direction.
3. Angle Between Vectors: The magnitude of the orthogonal projection is ||a|| * |cos(θ)|, where θ is the angle between a and b. If they are perpendicular (θ=90°), the projection is the zero vector. If they are parallel (θ=0° or 180°), the projection’s magnitude is ||a||.
4. Magnitude of Vector a: A longer vector a, at the same angle to b, will generally have a longer projection.
5. Magnitude of Vector b: While the magnitude of b appears in the denominator ||b||², it also appears in the final multiplication by b. The direction of the projection depends only on the direction of b, not its magnitude, but the calculation involves ||b||².
6. Zero Vector: You cannot project onto the zero vector (0,0) because ||b||² would be zero, leading to division by zero.

Frequently Asked Questions (FAQ)

Q1: What happens if vector b is the zero vector?
A1: The orthogonal projection onto the zero vector is undefined because it involves division by the squared magnitude of b, which would be zero. Our calculator will likely show NaN or infinity if you input (0,0) for vector b.

Q2: Is the orthogonal projection of a onto b the same as b onto a?
A2: No, generally proj_b a ≠ proj_a b unless a and b are parallel or one is the zero vector (where one is undefined). They are projections onto different lines/directions.

Q3: What is the difference between scalar projection and vector projection (orthogonal projection)?
A3: The scalar projection of a onto b is a scalar (a length), given by (a · b) / ||b||. The orthogonal projection (or vector projection) is a vector, found by multiplying the scalar projection by a unit vector in the direction of b: ((a · b) / ||b||²) * b.

Q4: Can the orthogonal projection be longer than the original vector a?
A4: No, the magnitude of the orthogonal projection of a onto b is ||a|| |cos(θ)|, and since |cos(θ)| ≤ 1, the magnitude of the projection is always less than or equal to the magnitude of a. It is equal only when a and b are parallel.

Q5: What if vectors a and b are perpendicular?
A5: If a and b are perpendicular, their dot product (a · b) is zero. Therefore, the orthogonal projection of a onto b is the zero vector (0,0).

Q6: How is orthogonal projection used in real life?
A6: It’s used to find the component of a force along a ramp (physics), in computer graphics to render shadows and transform coordinates, and in data analysis to reduce dimensionality (like in PCA).

Q7: What does the ‘orthogonal’ part mean?
A7: It refers to the fact that the line segment connecting the head of vector a to its projection onto the line of b is perpendicular (orthogonal) to vector b.

Q8: Can I use this calculator for 3D vectors?
A8: This specific calculator is designed for 2D vectors (a_x, a_y and b_x, b_y). For 3D vectors, you would add a z-component to each vector, and the dot product and magnitude calculations would include the z-components (a · b = a_xb_x + a_yb_y + a_zb_z and ||b||² = b_x² + b_y² + b_z²). The formula remains the same structure.