Vector Projection Calculator

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:

$$\mathbf{\vec{v}}=\langle2,3\rangle$$$$\mathbf{\vec{u}}=\langle1,1\rangle$$

The dot product is $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=2\cdot1+3\cdot1=5$$$.

The square of the magnitude of $$$\mathbf{\vec{u}}$$$ is $$$\mathbf{\left\lvert\vec{u}\right\rvert}^2=1^2+1^2=2$$$.

Therefore, the projection of $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$ is $$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)=\frac{\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert}^2}\mathbf{\vec{u}}=\frac{5}{2}\langle1,1\rangle=\langle\frac{5}{2},\frac{5}{2}\rangle$$$.

So the projection of the vector $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$ is the vector $$$\langle\frac{5}{2},\frac{5}{2}\rangle$$$, which has the same direction as $$$\mathbf{\vec{u}}$$$ and a magnitude equal to the component of $$$\mathbf{\vec{v}}$$$ in the direction of $$$\mathbf{\vec{u}}$$$.

How Is the Vector Projection Formula Derived?

Let's dive into the sequential process of deriving the vector projection formula.

• Step 1: Start with two vectors, $$$\mathbf{\vec{v}}$$$ and $$$\mathbf{\vec{u}}$$$.
• Step 2: Define the dot product of these two vectors, $$$\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}=\mathbf{\left\lvert\vec{u}\right\rvert}\mathbf{\left\lvert\vec{v}\right\rvert}\cos\left(\theta\right)$$$, where $$$\mathbf{\left\lvert\vec{u}\right\rvert}$$$ and $$$\mathbf{\left\lvert\vec{v}\right\rvert}$$$ are the magnitudes (or lengths) of the vectors $$$\mathbf{\vec{u}}$$$ and $$$\mathbf{\vec{v}}$$$ respectively, and $$$\theta$$$ is the angle between them.
• Step 3: Understand that the projection of the vector $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$ forms a right triangle with the vector $$$\mathbf{\vec{v}}$$$ and a vector parallel to $$$\mathbf{\vec{u}}$$$. In this right triangle, the hypotenuse is the vector $$$\mathbf{\vec{v}}$$$, the angle at the base is $$$\theta$$$, and the base represents $$$\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)$$$.

• Step 4: Apply the definition of cosine in the context of the right triangle you've just identified. The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. So

  $$\cos\left(\theta\right)=\frac{\left\lvert\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)\right\rvert}{\mathbf{\left\lvert\vec{v}\right\rvert}}$$

• Step 5: Substitute this definition of $$$\cos\left(\theta\right)$$$ into the dot product formula from step 2. This gives you:

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

• Step 6: Simplify the equation by canceling out the $$$\mathbf{\left\lvert\vec{v}\right\rvert}$$$ terms on the right side to get

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

• Step 7: Rearrange the equation to isolate $$$\left\lvert\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)\right\rvert$$$, the magnitude of the projection of $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$. This gives you

  $$\left\lvert\operatorname{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right)\right\rvert=\frac{\mathbf{\vec{u}}\cdot\mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert}}$$
• Step 8: Remember that you want the vector projection, not just its magnitude. A vector has both magnitude and direction. The projection of $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$ has the magnitude you've just found and the same direction as the vector $$$\mathbf{\vec{u}}$$$. The direction can be represented by the unit vector in the direction of $$$\mathbf{\vec{u}}$$$, i.e., $$$\frac{\mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}}$$$.

• Step 9: Combine the magnitude and direction to express the projection of $$$\mathbf{\vec{v}}$$$ onto $$$\mathbf{\vec{u}}$$$:

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

• Step 10: Simplify the equation to get the final formula for the vector projection

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

Why Choose Our Vector Projection Calculator?

• Accuracy

  Our calculator is carefully designed to deliver precise results, ensuring correct calculations every time.

• Ease of Use

  With a simple and intuitive interface, the process of calculating vector projections is quick and easy.

• Step-by-Step Explanations

  The calculator provides a comprehensive guide to each step of the calculation process, helping you understand how the vector projection is computed.

• Versatility

  Our calculator is not only useful for academic purposes in fields like mathematics or physics, but also for professionals working in areas such as engineering, computer graphics, and data analysis.

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