Vectors are represented by arrows indicating magnitude (length) and direction. Equivalent vectors have the same magnitude and direction regardless of starting point. Mathematically, vectors are denoted by lowercase bold letters (e.g., **a**) or with arrows (e.g., →a). In two dimensions, a vector is represented by an ordered pair (x, y), where x and y are the horizontal and vertical components respectively. The magnitude is calculated using the Pythagorean theorem. The concept extends to higher dimensions, though visualization becomes challenging beyond three. now, what's interesting about vectors is that we only care about the magnitude in the direction. we don't necessarily not care where we start, where we place it when we think about it visually like this and so we'll put our horizontal dimension first. so you might call this vector 5, 0. it's moving 5, positive 5 in the horizontal direction. and it's not moving at all in the vertical direction. and the notation might change. you might also see notation, and actually in the linear algebra context, it's more typical to write it as a column vector like this--5, 0. so this vector might be specified as 3, 4 [[## Linear Algebra: Introduction to Vectors Chapter 1: Introduction to Vectors This chapter introduces the fundamental concept of a vector, differentiating it from a scalar and providing a practical example. 1.1 Definition of a Vector 0]] A vector is a mathematical quantity that possesses both magnitude ( size or length ) and direction . 1.2 Vector vs. Scalar A scalar is a quantity that only has magnitude. A vector has both magnitude and direction. 1.3 Example: Speed vs. Velocity Speed : An example of a scalar quantity . If something is moving at 5 miles per hour, this only specifies magnitude. Velocity : An example of a vector quantity . If something is moving 5 miles per hour due east, this specifies both magnitude (5 mph) and direction (east). Chapter 2: Visualizing Vectors This chapter explains how vectors are visually represented, focusing on their key properties and the concept of equivalence. 2.1 Representing Vectors as Arrows Vectors can be visually represented as arrows , especially in two or three dimensions. The length of the arrow represents the vector's magnitude . The direction the arrow points represents the vector's direction . 2.2 Key Properties: Magnitude and Direction The length of the arrow signifies its magnitude (e.g., 5 units long for 5 miles per hour). The orientation of the arrow indicates its direction (e.g., pointing right for east). 2.3 Vector Equivalence An important characteristic of vectors is that they are defined solely by their magnitude and direction. Their starting point does not matter. Two vectors are considered equivalent if they have the exact same magnitude and direction, regardless of where they are placed. Chapter 3: Mathematical Notation for Vectors This chapter introduces the standard mathematical ways to represent vectors, particularly using component form. 3.1 Symbolic Representation When representing a vector as a variable: In print, it is often a lowercase letter in bold (e.g., ). In handwriting, it is typically a lowercase letter with an arrow on top (e.g., ). 3.2 Component Form in 2D A common way to represent vectors mathematically, especially for performing operations, is through their components. In two dimensions, a vector is specified by two numbers: one for its movement along the horizontal dimension and one for its movement along the vertical dimension. Example: A vector representing 5 miles per hour east, which only moves horizontally, can be written as . 3.3 Column Vector Notation In the context of linear algebra, it is more typical to write vectors as column vectors : The first coordinate represents movement in the horizontal direction. The second coordinate represents movement in the vertical direction. Example: for 5 units horizontal, 0 units vertical. Example: for 3 units horizontal, 4 units vertical. Chapter 4: Understanding Vector Components and Higher Dimensions This chapter delves into how vectors are broken down into their components, how their magnitude is calculated, and the utility of component notation for dimensions beyond what can be visualized. 4.1 Breaking Down Vectors A vector in component form, such as , indicates that it shifts 3 units in the horizontal direction and 4 units in the vertical direction from its start to its end. 4.2 Calculating Magnitude (Pythagorean Theorem) The magnitude (length) of a vector in component form can be calculated using the Pythagorean theorem . For a vector , its magnitude, often denoted as or , is given by the formula: $ Example: For the vector , the magnitude is: $ 4.3 Extending to Higher Dimensions While human brains struggle to visualize beyond three dimensions, the component notation allows for mathematical operations and understanding of vectors in four, five, or even more dimensions . This abstract representation is crucial for advanced linear algebra. Final Summary This introduction to vectors covered their fundamental definition as quantities with magnitude and direction , distinguishing them from scalars. We explored their visual representation as arrows , emphasizing that their position does not affect their identity (vector equivalence). Crucially, we learned about their mathematical notation in component form , particularly the column vector , which allows us to precisely define their horizontal and vertical movements. Finally, we saw how the Pythagorean theorem is used to calculate a vector's magnitude from its components and understood that this component notation is essential for working with vectors in higher dimensions that are beyond human visualization. be a vector, we would also have to specify the dir