Addition and Subtraction of Coplanar Vectors

Addition of Coplanar Vectors

Coplanar vectors are vectors that exist on the same plane.
Vector addition can occur geometrically or algebraically.
For geometric addition, place the initial point of the second vector at the terminal point of the first vector. The resultant vector is from the initial point of the first vector to the terminal point of the second vector.
For algebraic addition, add corresponding components of the two vectors.
The principle of triangle law states that the sum of two vectors is the third side of the triangle they form when arranged head-to-tail.
The parallelogram law of addition states that the sum of two vectors is the diagonal of the parallelogram they form when placed tail-to-tail.

Vector subtraction can be seen geometrically as vector addition in reverse.
For geometric subtraction, imagine reversing the direction of the subtracted vector and then performing vector addition.
Algebraic subtraction simply subtracts corresponding components of two vectors.
The triangle method for subtraction involves using the head-to-tail method, but for the vector being subtracted, reverse its direction.
Remember, the resultant vector will always start from the position of the first vector and ends at the position of the second vector.

Vector addition and subtraction obey the commutative law (changing the order does not change the result) and the associative law (grouping does not change the result).
Addition of a vector with its negation (same vector but in opposite direction) results in the zero vector.
Scalar multiplication affects the magnitude and direction of the vector but does not affect operations of addition and subtraction. It distributes over the operations (distributive law).

A scalar has magnitude only while a vector has both magnitude and direction.
Vectors can be equal if they have the same magnitude and direction, regardless of their initial and terminal points.
The magnitude of a vector is its length or size, denoted by A .
The direction of a vector is the way from its tail to head, usually given by an angle measure.
Vectors are denoted typically in bold or with an arrow above a letter.