Adding Vectors in Rectangular Form
Adding vectors in rectangular form is a fundamental mathematical process used to combine quantities that have both magnitude and direction. Vectors are represented as arrows, where the length indicates size and the direction denotes orientation. In two-dimensional space, a vector can be expressed as coordinates, for instance, v = (x, y), while in three-dimensional space, it extends to v = (x, y, z). The addition of vectors is accomplished by summing their respective coordinates; for example, if vector A is (xa, ya) and vector B is (xb, yb), the resultant vector R can be calculated as R = (xa + xb, ya + yb).
This method is visually represented as the hypotenuse of a right triangle formed by the two vectors, illustrating how they combine directionally. The process remains consistent regardless of the vectors' orientations, whether purely horizontal or vertical, or a combination thereof. A useful technique in vector addition is the tip-to-tail method, where the tail of one vector is placed at the tip of another. Additionally, vector addition is commutative, meaning the order of addition does not affect the final result. For vectors expressed in polar coordinates, conversion to rectangular form is often necessary before performing addition. Understanding these principles is essential for applications in physics where vectors commonly represent forces and velocities.
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Adding Vectors in Rectangular Form
Vectors are used to represent variables that have both size (magnitude) and direction. Combining such values is an important mathematical concept, and the main way of adding vectors is the use of rectangular form through coordinate values.
Vectors can be two-dimensional or three-dimensional and are represented in various ways. Graphically, a vector is shown as an arrow where the length of the arrow denotes the vector size and the orientation of the arrow denotes the direction of the vector. The ray from point A (xa, ya) to point B (xb, yb), vAB, can be thought of as a vector that is denoted 
or v = 
. The symbol for a vector is typically either bolded or shown with a ray above the label. The coordinates of the vector indicate where the vector is pointing, not necessarily its origin. This same logic follows in three-dimensional space where the vector v = (x, y, z).
Examples of variables that are frequently represented as vectors include forces and velocities. It is common for these values to be combined, so the addition of vectors is an important operation to understand and be able to do. The most common and most accurate method for adding vectors is in rectangular form. This corresponds to using the coordinate values to denote the vector and combining coordinates of like directions in the reference frame.
Overview
The concept of vectors is one that may have been established as early as Aristotle’s time, but the actual representation in a geometric form was not made until the early 1800s. Vectors can be added in rectangular form by adding coordinates in each direction. So if 
= (xa, ya) and 
= (xb, yb), the vector 
= (xa + xb, ya + yb).
A simple example of adding two vectors using rectangular form is the addition of a horizontal vector and a vertical vector. The addition of these gives a vector that contains both components and has a visual representation as the hypotenuse of a right triangle whose two legs are the vectors being added. The sum is the diagonal of the quadrilateral that is produced by the two vectors.
Most vectors are not purely horizontal or vertical, but the process is identical. Figure 1 illustrates the addition of two vectors. The resulting vector can be seen visually as the vector from the start of the first vector to the tail of the second vector, when it is drawn with its tail at the tip of the first vector (the gray dotted vector). This is often referred to as the tip-to-tail method.
Adding vectors is commutative, meaning that the order does not matter and that
. If vectors are represented in polar coordinates, it is easiest to convert to coordinate form, then combine the vectors. To convert from (r, () to (x, y), use x = rcos( and y = rsin(.
Bibliography
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