1 The Basics of Two-Dimensional Vectors

This chapter is designed to give anyone the basic idea of a vector, how they work, and how they interact with a cartesian plane. The most basic definition of a vector is an object with both direction and magnitude^[1]. This means a line in space with an angle relative to your 𝑥 and 𝑦 axis^[2], and a length to the line. A vector is drawn as an arrow, with the head pointing to the direction of the force.

The tail of the arrow represents the origin of the force. The magnitude, or the amount of force within the body of the arrow is relative to how long the arrow is drawn. The way I find best to picture vectors in my mind is by imagining them as bodies moving through space, away from an imaginary origin at the rate of the magnitude and at the direction set by the angle relative to the positive 𝑥 axis. This becomes useful when it comes time to combine vectors and to solve for a resultant, however the focus right now is understanding vectors.

Since the tip of the arrow represents the origin of the force and the length of the arrow is the amount of force within the drawn vector, one must account for the type of force acting upon the body. If the force is a compressive motion, such as pushing or hitting, the vector arrow should be drawn pointing to the body whilst maintaining the proper direction, as if it were pushing directly into the body. The contrary is drawn for tensile forces, such as pulling. When drawing a pulling force, you draw the tail of the arrow at the body and the head facing away from the body, as if it were pulling the body via a rope. This serves no benefit to calculations; all this does is provide an organized look to your vectors.

Vectors are used in science to describe anything that has direction and magnitude. Vectors can be used to describe nearly anything; a bird flying is a vector quantity. It has a direction (such as south) with a magnitude (30 kilometres per hour). You could map the vector of said bird by drawing the tip of the tail at the starting point of the bird’s journey and the head of the arrow at the end. One of my favourite examples of vectors are billiard balls; the ball’s velocity vector explains the movement of the ball, the length or magnitude of said vector is the speed the ball is traveling, and the direction of the vector is the ball’s angle of motion. Billiards balls come in handy when it comes time to study collisions in physics, however that includes initial mass, initial velocity and final of each. Such values we do not have, instead we derive a simple prediction^[3] based on the combination of selected vectors.

Vectors are also used when you have one or more opposing forces that will end up combined. For example, in hockey, if two players hit the puck at the same time with different magnitudes and different directions, where will the puck go? By using a vector system on a basic cartesian plane, we can ascertain an estimate of the endpoint of the puck, based on our given values.

In physics, vectors are used in a vast number of equations, expressed in force, drag velocity and acceleration to list a few. These vector quantities are essential in equations such as Newton’s second law, equations of motion, collisions, and gravitational fields. Vectors have allowed us to travel outside of our world, through the Apollo program and satellites. They also allow us to travel within it; vectors are used in aquatic construction allowing us to make boats and submersibles. Bridges can only be made with the help of vectors. Simply put, vectors are crucial for everything around us.