Vector Projection Calculator

Calculate vector projections step by step

The calculator will find the vector projection of one vector onto another, with steps shown.

Related calculator: Scalar Projection Calculator

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The Vector Projection Calculator is a cutting-edge instrument that performs vector projection computations with exceptional accuracy and precision. It provides a comprehensive step-by-step solution, serving as a reliable guide throughout the process.

How to Use the Vector Projection Calculator?

• Input

Input the coordinates of your vectors into the appropriate fields.

• Calculation

Once your vectors are correctly entered, click the "Calculate" button.

• Result

The calculator will quickly find the vector projection and present the resulting vector. For a better understanding, the calculator also delivers a comprehensive step-by-step guide that explains the entire calculation process.

What Is Vector Projection?

Vector projection is a significant concept in linear algebra and vector calculus. It refers to the process where one vector, often referred to as $\mathbf{\vec{v}}$, is projected onto another vector, referred to as $\mathbf{\vec{u}}$. The resultant vector, or the projection of $\mathbf{\vec{v}}$ onto $\mathbf{\vec{u}}$, has the same direction as $\mathbf{\vec{u}}$ and its length equals the component of $\mathbf{\vec{v}}$ that is in the same direction as $\mathbf{\vec{u}}$.

To better understand, let's delve into the formula for vector projection. Given two vectors $\mathbf{\vec{v}}$ and $\mathbf{\vec{u}}$, the projection of $\mathbf{\vec{v}}$ onto $\mathbf{\vec{u}}$ is calculated using the formula:

$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)=\frac{\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert}^2}\mathbf{\vec{u}},$$

where $\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}$ denotes the dot product of the vectors $\mathbf{\vec{u}}$ and $\mathbf{\vec{v}}$, which is the sum of the products of their corresponding coordinates; $\mathbf{\left\lvert\vec{u}\right\rvert}^2$ represents the square of the magnitude (or length) of the vector $\mathbf{\vec{u}}$.

The result, $\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)$, is the vector in the direction of $\mathbf{\vec{u}}$.

As an example, let's take two vectors:

How does the Vector Projection Calculator work?

The calculator operates by using the formula for vector projection, accurately calculating the projection of one vector onto another.