A vector lets you describe quantities in terms of an amount (called the magnitude) and a direction, making them a handy mathematical tool. Treating quantities as vectors opens up many powerful ways of calculating and analyzing forces, motion and other phenomena where direction plays a role. Vectors are indispensable not only in math itself, but also in hard sciences such as physics, and disciplines such as engineering. Though the math can be complex, the basic ideas behind vectors are not hard to grasp.
TL;DR (Too Long; Didn't Read)
TL;DR (Too Long; Didn't Read)
A vector is a quantify that has both an amount and a direction. Force and velocity are two examples of vector quantities.
Scalars and Vectors
Mathematicians call simple quantities scalars; these include properties such as temperature, weight and height, where a single number tells you everything you need. A vector also has an amount, but adds a direction; for example, a plane flies north at a speed of 645 kilometers per hour (400 miles per hour). The amount is the speed, 645 kph, and the direction is north. Both of these pieces of information form the plane’s velocity vector. Similarly, to open a door, you push on it with a force of 50 newtons (11 pounds). Fifty newtons is the magnitude; the direction is “away from the front of your body.” This forms the vector for the pushing force on the door.
It helps to visualize vectors by drawing them as arrows. The arrow points in the direction of the vector, and has a length that represents the vector’s magnitude. You can combine several vectors in the drawing, each with its own direction and length. In addition, you can choose between Cartesian (x and y) or polar coordinates (magnitude and angle). If your drawing skills are up to it, you can also sketch vectors in three dimensions using perspective and depth.
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Math with Vectors
Just as you can do math with scalar quantities, you can add and subtract vectors as well as perform other operations on them. One approach to adding vectors is to simply add up their x and y coordinates. For example, if you have two vector arrows, one of which has its tail at the origin, (0, 0), and head at (5, 5), and the other which also has its tail at the origin and has its head at (3, 0). Adding the x coordinates gives you 8, and adding the y locations gives 5, so the resulting vector is (8, 5).
Other operations with vectors include the dot product and the cross product; these are functions done in linear algebra that take two vectors and produce a result. The dot product yields a scalar that combines the lengths of the two original vectors. It applies to problems such as finding the energy needed to push a heavy object up a ramp. The cross product yields a third vector that points 90 degrees from either of the first two; it has applications in the forces of electricity and magnetism.
Physics, Engineering and Other Fields
It should come as no great surprise that you encounter vectors a great deal in physics and engineering. Vectors come in handy for solving problems involving quantities such as force, velocity and acceleration. Wind vectors help weather forecasters chart the progress of storms. These disciplines also make use of “vector fields,” or large groups of vectors spread out the represent phenomena such as the field lines around a magnet or the complex water currents in an ocean.