Find Projection Of U Onto V Calculator

Easily calculate the vector projection of vector u onto vector v using our free projection of u onto v calculator. Input the components of your 2D vectors and get the projection vector, dot product, and magnitude squared instantly. Understand the formula and visualize the result with our projection of u onto v calculator.

Vector Projection Calculator

Enter the components of vector u = (u₁, u₂) and vector v = (v₁, v₂):

Results Table

Vector	Component 1	Component 2
u	2	3
v	4	1
projvu	–	–

Table summarizing the components of vectors u, v, and the projection of u onto v.

What is the Projection of u onto v?

The projection of u onto v (denoted as proj_vu) is a vector that represents the "shadow" or component of vector u along the direction of vector v. Imagine a light source shining perpendicularly onto the line defined by vector v; the shadow cast by vector u onto that line is the projection of u onto v. It's a fundamental concept in linear algebra and physics, used in various applications like finding the component of a force along a certain direction or in computer graphics.

The result of the projection is always a vector that is parallel to v (or is the zero vector if u and v are orthogonal or if u is the zero vector). Our projection of u onto v calculator helps you find this resulting vector quickly.

Anyone working with vectors, such as students of physics, engineering, mathematics, computer graphics developers, and data scientists, might need to use a projection of u onto v calculator.

A common misconception is that the projection is a scalar (a single number), but it is actually a vector. The scalar multiple that scales v to get the projection is indeed a scalar, but the projection itself is a vector having both magnitude and direction.

Projection of u onto v Formula and Mathematical Explanation

The formula to calculate the vector projection of u onto v is:

proj_vu = [(u · v) / |v|²] * v

Let's break it down:

1. u · v: This is the dot product (or scalar product) of vectors u and v. If u = (u₁, u₂) and v = (v₁, v₂), then u · v = u₁v₁ + u₂v₂. The dot product is a scalar.
2. |v|²: This is the squared magnitude (or squared length) of vector v. If v = (v₁, v₂), then |v| = √(v₁² + v₂²), so |v|² = v₁² + v₂². This is also a scalar, and it must be non-zero (i.e., v cannot be the zero vector).
3. (u · v) / |v|²: This is a scalar value that represents how much vector v needs to be scaled to get the projection.
4. [(u · v) / |v|²] * v: The scalar value is then multiplied by vector v, scaling v to produce the projection vector, which lies along the direction of v.

The projection of u onto v calculator performs these steps automatically.

Variables Table

Variable	Meaning	Unit	Typical Range
u1, u2	Components of vector u	Depends on context (e.g., meters, m/s)	Any real number
v1, v2	Components of vector v	Same as u	Any real number (v is not zero vector)
u · v	Dot product of u and v	(Unit of u) * (Unit of v)	Any real number
|v|^2	Squared magnitude of v	(Unit of v)^2	Positive real number
projvu	Projection vector of u onto v	Same as u and v	Vector components are real numbers

Variables involved in the projection of u onto v calculation.

Practical Examples (Real-World Use Cases)

Example 1: Force Component

Imagine a force vector F = (10, 5) N (Newtons) acting on an object, and we want to find the component of this force along the direction of a ramp defined by the vector d = (4, 3).

• u = F = (10, 5)
• v = d = (4, 3)
• F · d = (10 * 4) + (5 * 3) = 40 + 15 = 55
• |d|² = 4² + 3² = 16 + 9 = 25
• Scalar = 55 / 25 = 2.2
• Projection of F onto d = 2.2 * (4, 3) = (8.8, 6.6) N

The component of the force along the ramp is (8.8, 6.6) N. You can verify this using our projection of u onto v calculator by setting u1=10, u2=5, v1=4, v2=3.

Example 2: Work Done

In physics, the work done by a constant force F moving an object along a displacement vector d is W = F · d. This is also related to the projection: W = |proj_dF| * |d| (if projection and d are in the same direction). Let F = (3, 4) N and d = (5, 0) m.

• u = F = (3, 4)
• v = d = (5, 0)
• F · d = (3 * 5) + (4 * 0) = 15
• |d|² = 5² + 0² = 25
• Scalar = 15 / 25 = 0.6
• Projection of F onto d = 0.6 * (5, 0) = (3, 0) N

The projection of the force onto the displacement is (3, 0) N. The work done is 15 Joules. Using the projection of u onto v calculator with u1=3, u2=4, v1=5, v2=0 gives proj=(3,0).

How to Use This Projection of u onto v Calculator

1. Enter Vector u Components: Input the values for u₁ and u₂ into the respective fields.
2. Enter Vector v Components: Input the values for v₁ and v₂. Make sure that vector v is not the zero vector (0,0), as projection onto the zero vector is undefined.
3. View Results: The calculator automatically updates the "Results" section, showing the primary result (the projection vector), the dot product, the squared magnitude of v, and the scalar multiple.
4. See Visualization: The chart below the calculator visualizes vectors u, v, and the projection.
5. Check Table: The table summarizes the components of all three vectors.
6. Reset: Click the "Reset" button to return to the default values.
7. Copy Results: Click "Copy Results" to copy the input and output values to your clipboard.

The projection of u onto v calculator provides immediate feedback, making it easy to understand how changes in the input vectors affect the projection.

Key Factors That Affect Projection of u onto v Results

• Components of u: Changing the components of u changes the vector being projected, directly affecting the dot product and thus the final projection.
• Components of v: Changing v alters the direction onto which u is projected and also the magnitude of v, affecting both the dot product and |v|². Vector v cannot be the zero vector.
• Angle Between u and v: The dot product u · v = |u||v|cos(θ), where θ is the angle between u and v. The projection's magnitude depends on cos(θ). If θ=90° (orthogonal), the projection is the zero vector. If θ=0° or 180° (parallel), the projection's magnitude is |u|.
• Magnitude of v: While the direction of the projection depends only on the direction of v, its length is inversely proportional to |v|² after being scaled by u · v. However, the projection vector itself scales proportionally with v.
• Zero Vector: If v is the zero vector, the projection is undefined because division by zero (|v|²=0) occurs. The projection of u onto v calculator handles this.
• Dimensionality: While this calculator is for 2D vectors, the concept extends to 3D and higher dimensions. For 3D vectors u=(u1,u2,u3) and v=(v1,v2,v3), u·v = u1v1+u2v2+u3v3 and |v|^2 = v1^2+v2^2+v3^2, but the formula remains the same.

Frequently Asked Questions (FAQ)

What is the projection of u onto v if u and v are orthogonal?

If u and v are orthogonal (perpendicular), their dot product (u · v) is zero. Therefore, the projection of u onto v is the zero vector (0, 0).

What is the projection of u onto v if v is the zero vector?

The projection of u onto the zero vector is undefined because it involves division by the squared magnitude of v, which would be zero. Our projection of u onto v calculator will indicate this.

Is the projection of u onto v the same as the projection of v onto u?

No, not generally. Proj_vu is parallel to v, while proj_uv is parallel to u. They are only the same if u and v are parallel or one is the zero vector (and the other is not, for proj_uv).

Can the projection vector be longer than vector u?

Yes. If the angle between u and v is small and |v| is small compared to |u|cos(θ), the scalar multiple can be large. However, the magnitude of the projection is |u||cos(θ)|, which is always less than or equal to |u|.

Does the order of vectors matter in the dot product?

No, the dot product is commutative: u · v = v · u.

What if I have 3D vectors?

The formula is analogous. If u=(u1,u2,u3) and v=(v1,v2,v3), then u·v = u1v1+u2v2+u3v3 and |v|^2 = v1^2+v2^2+v3^2. The projection is still [(u·v)/|v|^2]*v. This projection of u onto v calculator is for 2D vectors, but the principle is the same.

What does the scalar multiple represent?

The scalar multiple [(u · v) / |v|²] tells you how many times you need to scale vector v to get the projection vector. It's the ratio of the projected length along v to the length of v, considering direction.

How is the projection related to the component of u along v?

The scalar component of u along v is (u · v) / |v|. The projection is this scalar component multiplied by the unit vector in the direction of v, which is v/|v|. So, proj_vu = [(u · v) / |v|] * (v/|v|) = [(u · v) / |v|²] * v.