In physics, engineering, and mathematics, a vector is a quantity that has both magnitude (size) and direction. Unlike scalar quantities (like temperature or mass) which only have magnitude, vectors are crucial for describing phenomena such as force, velocity, acceleration, and displacement. A common way to represent a 2D vector is by its components along the X and Y axes, written as (x, y).
Components of a Vector
Every 2D vector can be broken down into two perpendicular components: an X-component and a Y-component. For example, a vector A can be written as A = (Ax, Ay), where Ax is its projection along the X-axis and Ay is its projection along the Y-axis. These components tell us how much the vector extends in each direction from its starting point.
Vector Addition
When you add two vectors, you are essentially combining their effects. Geometrically, if you place the tail of the second vector at the head of the first, the resultant vector (sum) goes from the tail of the first to the head of the second. Mathematically, vector addition is performed by adding their corresponding components:
If A = (Ax, Ay) and B = (Bx, By), then A + B = (Ax + Bx, Ay + By).
Example: If A = (3, 4) and B = (1, 2), then A + B = (3+1, 4+2) = (4, 6).
Vector Subtraction
Vector subtraction can be thought of as adding the negative of a vector. The negative of a vector B, denoted as –B, has the same magnitude as B but points in the opposite direction. Mathematically, vector subtraction is performed by subtracting their corresponding components:
If A = (Ax, Ay) and B = (Bx, By), then A – B = (Ax – Bx, Ay – By).
Example: If A = (3, 4) and B = (1, 2), then A – B = (3-1, 4-2) = (2, 2).
Dot Product (Scalar Product)
The dot product is an operation that takes two vectors and returns a single scalar number. It is a measure of how much two vectors point in the same direction. If the dot product is positive, the vectors generally point in the same direction; if negative, they generally point in opposite directions; if zero, they are perpendicular.
For 2D vectors A = (Ax, Ay) and B = (Bx, By), the dot product is calculated as:
A · B = AxBx + AyBy
Example: If A = (3, 4) and B = (1, 2), then A · B = (3 * 1) + (4 * 2) = 3 + 8 = 11.
Magnitude of a Vector
The magnitude of a vector represents its length or size. It is a scalar quantity. For a 2D vector, its magnitude can be calculated using the Pythagorean theorem, as the components form the sides of a right-angled triangle.
For a vector A = (Ax, Ay), its magnitude (|A|) is:
|A| = √(Ax2 + Ay2)
Example: If A = (3, 4), then |A| = √(32 + 42) = √(9 + 16) = √25 = 5.
These fundamental operations are essential tools in various scientific and engineering disciplines for analyzing forces, motion, and spatial relationships.