Introduction to Vectors in Physics
Vectors are a simple approach to solving many classical problems, including dynamics, ballistics, and problems dealing with gravitation. In fact, vectors are so integral to physics that most courses begin with instruction on their use. Thus it is essential that any physics student has a firm grasp on the concepts presented here.
Section 1.0 – The Scalar Value
A scalar is any data that has only magnitude (an amount, or a number), but not direction. Thus it is a one-dimensional unit of data. Notable scalars include speed, distance, time, mass, and luminosity.
Section 2.0 – The Vector Value
A vector is a two-dimentional piece of data, including both magnitude and direction components. Although more than one piece of information is contained within a vector, it is still considered as one unit of data. Because of their two-dimensional nature, vectors can be effectively represented on a euclidean plane. The direction is represented as an angle.
Section 3.0 – Adding Simple Vectors
As vectors can be described on a euclidean plane, the distance between two vectors can be calculated using the formula for the distance between two points, where the points are the endpoints of the vectors. Therefore, a combination of trigonometry and basic geometry can be used to solve for the endpoints of the vector, which can then serve in the equation.
Section 4.0 – Components of Vectors
Any vector can be broken into two components: a horizontal (x) component, as well as a vertical (y) component. If the magnitude (d) and angle (a) of a vector is known, the horizontal component of the vector is:
x = d cos a
The vertical (y) component is then:
y = d sin a
Section 5.0 – Sample Problem
Force A is applied to a ball at 50 degrees with a magnitude of 50N. Force B is applied with a magnitude of 20N at 120 degrees. What is the magnitude of the resulting force?
To solve this problem, simply split the two vectors into their components.
Xa = 50 cos 50
Ya = 50 sin 50
Xb = 20 cos 120
Yb = 20 sin 120
The resultant vector is simply equal to the sum of the components:
Rx = Xa + Xb
Ry = Ya + Yb
Therefore, the points (Xa, Ya) and (Xb, Yb) are substituted into the distance formula, yielding the result of 36.5N
These are the basics of vectors in physics. A good knowledge of vectors is essential to the success of any physics student. Good luck, and happy calculating!