What are vectors?
By definition, a vector is an object (represented as an arrow) that always has magnitude (size) and direction. When you draw a vector (as an arrow), the length of the vector represents the magnitude i.e. the size in a particular direction.
Vectors can be extrapolated to more dimensions such as 3D vectors, but we will be focusing on 2D vectors. Since we are working 2D vectors, a vector can be broken down into two components: x and y (horizontal and vertical) are represented as (x,y). A vector is also written with a little arrow above the letter symbolizing it like so: . Take a look at an example of a vector below – the vector has a magnitude, a direction, a horizontal and vertical component. We say this vector =x,y=(3,2).
How to use this concept:
The concept of vectors can go in-depth a lot as there are a lot of things you can do with them.
Magnitude and Direction
One of the most foundational ideas of vectors is that each vector has a magnitude and direction. The magnitude is always represented by two vertical bars on either side of the vector (or double vertical bars) like so:
The magnitude of a vector is a number/scalar indicating the size of the vector and can be calculated using the Pythagorean theorem:
On the other hand, the direction is just a measure of an angle made between the vector placed starting from (0,0) on the cartesian plane and the positive x-axis. It can be calculated by using simple trigonometry with tangent:
Addition and Subtraction
Another foundational idea of vectors is vector addition and subtraction. Vector addition is pretty straight-forward. When you add two vectors, you get a resultant vector whose x and y components are just a sum of the x and y components of the two smaller vectors – have a look at the diagram below:
When it comes to subtraction, the same principle applies except the negative is absorbed by the x and y components ea
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More: Vectors don’t stop here! There are more vector operations, specifically vector by vector products such as dot and cross product.
Sample Math Problems
Question 1:
Find the magnitude and direction angle for the vector =(-8, 41)
Answer:
To find the magnitude, we just need to apply its formula that resembles the Pythagorean theorem:
Now to find the direction, we need to find the angle using trigonometry:
Note, when we do arctan(), the angle that we get back is considered as the principle/critical angle. Since the range of arctan lies only from -pi/2 to pi/2, the output we get by doing arctan() only gives the angle either in the first or fourth quadrant. But if we were to actually visualize where our vector would be, it would lie in the second quadrant. Then appropriately, we need to change our angle by the quadrant to get the actual angle of the vector:
