The term “vector" is used to describe those physical quantities that cannot be completely described just by their magnitudes; Rather these physical quantities need to be described by their magnitudes as well as their direction.
For example, speed is a scalar quantity; We can say speed of an object is 10 m/s, whereas velocity is a vector quantity, and to describe the velocity of an object, we can say the velocity of a car is 10 m/s in the North direction.How can a vector be represented on a graphA vector quantity is represented on a graph by an arrow.
The length of the arrow represents the magnitude of the vector, whereas the arrow head represents the direction of a vector. The following diagram represents a vector of magnitude 10 and direction East. We assume a scale of 10 : 2 cm on the graph:Representation of a vectorHow can a vector be resolved into componentsA vector can be represented as the resultant of two or three vectors in a two or three dimensional graph respectively.
These parts of that vector are called its components. In the diagram below, the vector of `vecA` is resolved into two components, vectors `vec(Ax)` and `vec(Ay)` :Components of a vectorRectangular components of a vector in physicsAs stated above, a vector can be resolved into two or three or more components. When a vector on a two dimensional graph is resolved into two components such that they are parellel to the two axes (the x-axis and the y-axis), the two components of that vector are called its rectangular components.
The rectangular components of a vector on a two dimensional graph.Similarly, when a vector is resolved into three components on a three dimensional graph, where each component is parallel to each axis (x, y and z axes), these components are also referred to as rectangular components. The below diagram shows the rectangular components of a vector on a three dimensional graph:Rectangular components of a vector on a three dimensional , the vector `vecA` is resolved into three rectangular components `vec(Ax)` , `vec(Ay)` and `vec(Az)` along the x, y and z axis respectively. On other words, vector `vecA` is the resultant of vectors `vec(Ax)` , `vec(Ay)` and `vec(Az)` . Thus,`vecA = vec(Ax) + vec(Ay) + vec(z)`
Formula for rectangular components of a vector in physicsIn the above diagram, the unit vectors along each axis (x, y and z axes) are `hati` , `hatj` and `hatk` respectively.We know that the magnitude of rectangular components of `vecA` is`|vec(Ax)| = Ax``|vec(Ay)| = Ay``|vec(Az)| = Az`Thus, the rectangular components of `vecA` can also be written as`vec(Ax) = Ax*hati``vec(Ay) = Ay*hatj``vec(Az) = Az*hatk`Thus, the vector `vecA` can be represented as a sum of the three rectangular components as follows:-`vecA = Ax*hati + Ay*hatj + Az*hatk`