Vector Projection Calculator

How to Calculate Vector Projection

Vector projection is a fundamental concept in linear algebra and vector mathematics. It involves finding the component of one vector that is parallel to another vector. This guide will explain how to calculate vector projection step by step.

Vector Projection Formulas

For vectors \(\vec{a}\) and \(\vec{b}\), the projection of \(\vec{a}\) onto \(\vec{b}\) is defined as follows:

• Scalar Projection: \(proj_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\)
• Vector Projection: \(proj_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b}\)

Where:

• \(\vec{a} \cdot \vec{b}\) is the dot product of vectors \(\vec{a}\) and \(\vec{b}\)
• |\(\vec{b}\)| is the magnitude (length) of vector \(\vec{b}\)

Calculation Steps

1. Calculate the dot product of vectors \(\vec{a}\) and \(\vec{b}\).
2. Calculate the magnitude of vector \(\vec{b}\).
3. Divide the dot product by the magnitude of \(\vec{b}\) to get the scalar projection.
4. Multiply the scalar projection by the unit vector in the direction of \(\vec{b}\) to get the vector projection.

Example Calculation

Let's calculate the projection of \(\vec{a} = (3, 4)\) onto \(\vec{b} = (1, 2)\):

Step 1: Calculate dot product

\(\vec{a} \cdot \vec{b} = (3 * 1) + (4 * 2) = 3 + 8 = 11\)

Step 2: Calculate magnitude of \(\vec{b}\)

|\(\vec{b}\)| = \(\sqrt{1^2 + 2^2} = \sqrt{5}\)

Step 3: Calculate scalar projection

Scalar Projection = \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{11}{\sqrt{5}} \approx 4.919\)

Step 4: Calculate vector projection

Vector Projection = \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} = \frac{11}{5} * (1, 2) \approx (2.2, 4.4)\)

Visual Representation

This diagram illustrates the vector projection. The blue arrow represents vector \(\vec{a}\), the red arrow represents vector \(\vec{b}\), and the green arrow shows the projection of \(\vec{a}\) onto \(\vec{b}\).