In this lesson, we will explore the world of Euclidean vectors. Vectors are a fundamental concept in mathematics and geometry, used to represent quantities that have both magnitude and direction. Understanding vectors is critical for various applications in physics, engineering, computer graphics, and more.
A Euclidean vector, often simply called a vector, is a geometric object that has magnitude (or length) and direction. Vectors can be represented in multiple dimensions, but for simplicity, we will begin with vectors in a two-dimensional space. A vector is typically drawn as an arrow, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow shows the direction of the vector.
Vectors can be represented in several ways. One common representation is in the form of coordinates, such as \( \vec{v} = (x, y) \), where \(x\) and \(y\) are components of the vector in the horizontal and vertical directions, respectively. Another way to represent vectors is using the notation \( \vec{v} = x\hat{i} + y\hat{j} \), where \(\hat{i}\) and \(\hat{j}\) are unit vectors in the horizontal and vertical directions, respectively.
The magnitude of a vector \(\vec{v} = (x, y)\) can be calculated using the Pythagorean theorem: \(|\vec{v}| = \sqrt{x^2 + y^2}\). The direction of a vector is often described using the angle \(\theta\) it makes with the positive x-axis, which can be determined using trigonometry, specifically via the tangent function: \(\tan(\theta) = \frac{y}{x}\).
There are several fundamental operations that can be performed with vectors, including addition, subtraction, and scalar multiplication.
The dot product (or scalar product) of two vectors \(\vec{a}\) and \(\vec{b}\), denoted as \(\vec{a} \cdot \vec{b}\), is a way of multiplying vectors to obtain a scalar (number). The dot product is defined as \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \), where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). If the vectors are given by their components, \(\vec{a} = (a_x, a_y)\) and \(\vec{b} = (b_x, b_y)\), the dot product can be calculated as \( \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y \). The dot product can be used to determine the angle between two vectors or to project one vector onto another.
The projection of vector \(\vec{a}\) onto vector \(\vec{b}\) is a vector that represents the component of \(\vec{a}\) in the direction of \(\vec{b}\). The projection is given by the formula: \( proj_{\vec{b}}\vec{a} = \left( \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \right) \vec{b} \), which scales \(\vec{b}\) according to the dot product ratio.
In three-dimensional space, another operation called the cross product is used to find a vector perpendicular to two other vectors. If \(\vec{a}\) and \(\vec{b}\) are vectors in 3D space, the cross product \(\vec{a} \times \vec{b}\) is a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) whose magnitude is \(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta)\), and \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). This operation is crucial in physics and engineering for finding normal vectors to surfaces or in calculating torque.
Vectors find applications in numerous fields:
To visualize vector addition, consider two vectors \(\vec{a} = (3, 2)\) and \(\vec{b} = (2, 1)\). To add these vectors, draw \(\vec{a}\) starting from the origin (0, 0) and then draw \(\vec{b}\) starting from the head of \(\vec{a}\). The resultant vector \(\vec{r}\) goes from the origin to the head of \(\vec{b}\) which was placed at the head of \(\vec{a}\). The coordinates of \(\vec{r}\) can be calculated as \(\vec{r} = \vec{a} + \vec{b} = (3+2, 2+1) = (5, 3)\).
Euclidean vectors are an essential part of mathematics and geometry, providing a way to quantify and manipulate quantities that have both magnitude and direction. Through operations like addition, subtraction, dot product, and cross product, vectors can model real-world phenomena and solve complex problems in various fields. By understanding the basic principles of vectors, you can build a foundation for more advanced studies in physics, engineering, computer science, and other disciplines. The geometric intuition and algebraic operations introduced with vectors pave the way for more complex concepts like vector spaces and linear transformations, which play a crucial role in higher mathematics and practical applications alike.
