Vector Projection Calculator

What Is The Vector Projection?

In physical terms:

“The length of the shadow that a vector makes over another vector is known as the vector protection”

Vector Projection Formula:

You can determine the vector projection of a vector onto another using the formula:

\( \text{Vector Projection: } \text{proj}_{\vec{u}} \vec{v} = \frac{\vec{v} \cdot \vec{u}}{|\vec{u}|^2} \, \vec{u} \)

Our free projection calculator uses this formula to calculate the resultant vector and its magnitude along another vector.

Scalar Projection:

The scalar projection (also called the component of one vector along another) is calculated as:

\( \text{Scalar Projection: } |\text{proj}_{\vec{u}} \vec{v}| = \frac{\vec{v} \cdot \vec{u}}{|\vec{u}|} \)

How To Find The Vector Projection?

Using this vector projection formula, you can compute the component of one vector along another. For practical understanding, check the examples below and try using the vector projection calculator for instant results.

Example #01:

Find the projection of the vector \(\vec{A}\) onto the vector \(\vec{B}\) with the following values:

\( \vec{A} = 6\hat{i} + 3\hat{j} + 8\hat{k} \)

\( \vec{B} = 3\hat{i} + 7\hat{j} + 1\hat{k} \)

Solution:

The formula for vector projection is:

\( \text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \, \vec{B} \)

Step 1: Compute the dot product of \(\vec{A}\) and \(\vec{B}\):

\( \vec{A} \cdot \vec{B} = (6)(3) + (3)(7) + (8)(1) = 18 + 21 + 8 = 47 \) (You can also use a dot product calculator)

Step 2: Compute the magnitude squared of \(\vec{B}\):

\( |\vec{B}|^2 = 3^2 + 7^2 + 1^2 = 9 + 49 + 1 = 59 \) (You can also use a vector magnitude calculator)

Step 3: Apply the vector projection formula:

\( \text{proj}_{\vec{B}} \vec{A} = \frac{47}{59} \, (3, 7, 1) \)

\( \text{proj}_{\vec{B}} \vec{A} = \left( \frac{141}{59}, \frac{329}{59}, \frac{47}{59} \right) \)

This is the required vector projection, which can also be calculated using our vector projection calculator.

How Vector Projection Calculator Works?

Let’s have a look at how this best projection of u onto v calculator will let you calculate the projection of one vector onto the other vector in a blink of moments:

Input:

• First of all, select whether you want to do calculations in the 2D system or 3D system from the first list
• After you do that, you need to select whether you will be selecting coordinates or points for first vector from second drop down list
• Now it's time to put the values of the first vector in their designated fields
• Repeat the same for the second vector
• Once you are done completely, hit the calculate button

Output: The best projection vector calculator calculates:

• Vector projection
• Scalar projection

FAQ’s:

What is the projection rule?

Projection law states that:

“If we calculate the sum of the two sides of the triangle made by vector combination, it would actually be equal to the third side”

What is orthogonal projection?

It is the means of displaying 3D objects in space as 2D objects. Being a special form of the parallel projection, it shows lines that are exactly at the right angle to that of the projection plane. This free orthogonal projection calculator will also let you determine such projection of vectors in a blink of moments.

What is the difference between 2D and 3D projection vectors?

2D vectors form a flat image which contains only two coordinates x and y. While on the other hand, the 3D vector introduces another coordinate Z that helps you to go in more depth understanding about the concept.

Why do we use orthographic projection?

Orthographic projection allows you to analyse each and every point drawn in the vector drawing. Moreover, it also assists you in determining the technical information about the vector sheets. And this free orthogonal projection matrix calculator will let you estimate the right angles used in the drawings to reduce error chances.

Conclusion:

Vector projection is a technique in physics that assists you in calculating missing components of the vector in a rectangular coordinate system. It is very handy in simplifying physical concepts like force and work to understand them better. And as far as the fast computations are concerned, this best vector projection calculator will always let you do so without any hurdles.

References:

From the source of Wikipedia: Vector projection, Notation, Scalar projection, Vector projection, Scalar rejection, Vector rejection, Properties From the source of Khan Academy: Projection of a vector on a line, Dot product From the source of Lumen Learning: Coordinate Systems, Polar Coordinates, Vectors in Three Dimensions