Vector Addition, Subtraction, and Geometrical Applications

Vectors are mathematical entities used to represent quantities with both magnitude and direction. In this article, we will explore how vectors can be added together, subtracted from one another, and applied within geometrical shapes like triangles and parallelograms.

Vector Addition

Vector addition is a process where two vectors are combined to form a single resultant vector. This operation follows some commonly known rules:

1. Commutativity: ( \vec{a} + \vec{b} = \vec{b} + \vec{a} )
2. Associativity: ( (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}) )

The sum of their magnitudes must equal the magnitude of the resultant vector, while the angle between them determines its direction. However, it's important to note that simple addition is only applicable when the vectors being added have the same starting point (tail), which may be different if they were initially drawn separately.

Vector Subtraction

Just as with regular numbers, you can subtract one vector from another using the following formula: [ \vec{a}-\vec{b}=\vec{a}+(-\vec{b}). ] Here, -(\vec b) represents a new vector that has the opposite direction of (\vec{b}) and the same magnitude.

This rule allows us to perform operations such as moving along (\vec{a}) until we reach the endpoint of (\vec{b}), effectively taking away (\vec{b}) from our path along (\vec{a}).

Vector Addition in Triangle Method

In geometry, especially when working with triangles or other polygons, there might arise situations where vectors need to be represented by line segments instead of arrows. To find the total displacement in these cases, we follow the steps outlined below:

1. Draw all legs of the figure.
2. Superimpose the initial position of each leg onto the tail of the next leg.
3. Use vector addition to determine the final position.

For instance, consider adding three vectors A, B, and C. First, draw A, superimpose B on top of A, then C on top of AB. Finally, calculate the net displacement by drawing the resultant vector AC.

Parallelogram Method

When dealing with quadrilaterals rather than just triangles, we often encounter objects called parallelograms. These are unique geometric figures defined by their four sides, all parallel to each other. For example, a square is simply a parallelogram with four congruent sides. When finding the area inside such a shape, we use what's called the parallelogram method:

1. Divide the parallelogram into two non-overlapping smaller parallelograms that share one common side.
2. Calculate the areas of these smaller parallelograms.
3. Sum up the areas so calculated.

Let's assume our parallelogram has side lengths AB=AC and BC=CD, and let P and Q be arbitrary points on AB and CD respectively. Then, we can easily see that the squares APBQ and CPDQ represent our original parallelogram. The area enclosed by any parallelogram equals half the product of its height and base length. Thus, we can compute the area of a parallelogram as ½ * height * base.

Explore the rules and methods of adding and subtracting vectors, as well as their applications in geometrical shapes like triangles and parallelograms. Learn about vector operations, commutativity, associativity, and how vectors can be used to find displacements and areas in geometric figures.