Calculator Online

Home AI Chat AI Math Solver

NEW

Tools ▾ Category ▾

Pricing Sign In

Calculator Online

Your Result is copied!

Vector Projection Calculator

Select the dimension and representation, then enter the required coordinates. The calculator will determine the projection of one vector onto another, displaying step-by-step calculations.

Dimensions of a vector

First Vector (A) Representation

First Vector (a)

Initial point:

Second Vector:

First Vector (B) Representation

Terminal point (b)

Initial point (A):

Second Vector (B):

Related

Get an instant calculation for the projection of one vector onto another with this free vector projection calculator. Determining the shadow of a vector on another vector is now just a few clicks away. Read on to understand this physical quantity and how to measure it efficiently.

What is Vector Projection?

In simple terms:

The vector projection represents the length of the shadow that one vector casts onto another vector.

vector projection

Vector Projection Formula

The projection of vector v onto vector u is calculated as:

proj_u v = (v · u / ||u||²) u

The calculator uses this formula to determine the resulting vector, which represents the “shadow” of one vector along another.

Scalar Projection

The scalar projection (component of v along u) is:

|proj_u v| = (v · u) / ||u||

How to Find Vector Projection

The vector projection calculator provides instant results. Here’s an example:

Example #1

Find the projection of vector A onto vector B:

A = 6î + 3ĵ + 8k̂

B = 3î + 7ĵ + 1k̂

Solution

Step 1: Calculate the dot product:

A · B = 6×3 + 3×7 + 8×1 = 47 (Use Dot Product Calculator for detailed calculation)

Step 2: Find the magnitude of B:

||B|| = √(3² + 7² + 1²) = √59 (Use Vector Magnitude Calculator)

Step 3: Apply the vector projection formula:

proj_B A = (47 / (√59)²) (3, 7, 1) = (47 / 59) (3, 7, 1)

Vector Projection = (141/59, 329/59, 47/59)

How the Vector Projection Calculator Works

Input:

Select 2D or 3D vector system
Choose input method: coordinates or points
Enter values for the first vector
Enter values for the second vector
Click Calculate

Output:

Vector projection (proj_B A)
Scalar projection (component of A along B)

FAQ

What is the projection rule?

The projection rule states: “The sum of the two sides of a triangle formed by vector combination equals the third side.”

What is orthogonal projection?

Orthogonal projection represents 3D objects as 2D objects with lines perpendicular to the projection plane. Use our orthogonal projection calculator for accurate calculations.

Difference between 2D and 3D vector projections

2D vectors include x and y coordinates only, while 3D vectors add a z-coordinate, providing full spatial information.

Why use orthographic projection?

Orthographic projection allows precise analysis of vector drawings, ensures accurate representation, and reduces errors. Use the orthogonal projection matrix calculator for estimating right angles in technical drawings.

Conclusion

Vector projection is useful in physics and engineering to calculate components of a vector along another. It simplifies concepts such as force and work. With this fast vector projection calculator, accurate calculations are performed instantly.

References

Sources: Wikipedia – Vector Projection; Khan Academy – Dot Product; Lumen Learning – Coordinate Systems & Vectors in 3D.

Related

Calculator Online

Get the ease of calculating anything from the source of calculator online

Links

Home Conversion Calculator About Calculator Online Blog Hire Us Knowledge Base Sitemap Sitemap Two

Support

Email us at