In today’s video, we will be introducing scalars and vectors, as well as different vector forms, vector operations, and unit vectors.
Scalars are just regular old numbers, like time (3PM) or distance (5 kilometers).
Vectors have a number associated with them, but also a direction, like displacement (5 km [North]) or acceleration (6 meters/second [down]).
Vectors look like this – they are lines with arrowheads at the end. The arrow end is the “tip”, the back end is the “tail.” The direction that the arrowhead points is the direction of your vector – which can be North, South, East, West, up, down, left, right, or on angles like North 30 degrees East, West 51 degrees South, and so on.
Vector Operations
There are different ways to combine or manipulate vectors. These include:
(a) Addition
Adding two vectors, which can be in 1D, 2D or 3D.
(b) Scalar Multiplication
Multiplying a vector by a constant number, like 4 or 1/2 (a scalar value)
(c) Dot Product
It gives us a scalar value as an answer, that tells us how much of one vector is in the direction of the other. This is the equation:
(d) Cross Product
It gives us a vector as an answer, and that vector is perpendicular (orthogonal) to the two vectors we took the cross product of. The method of solving the cross product is discussed in the video.
Vector Notations
There are different ways of writing out vectors. These forms are explained in the video, but I will give a brief overview here:
(a) Magnitude Direction Form
You need to find the magnitude of the vector, along with the angle that it makes with the x-axis.
(b) Component Form
You need to know the coordinate points of your vector.
(c) Unit Vector Form
Same as (b). You need to know the coordinate points of your vector.
Unit Vectors
A unit vector is just a vector with a magnitude of 1. This means that if I use the following equation to find the magnitude of a vector:
Then I would get 1 if it is indeed a unit vector! We have 3 basic unit vectors, that are of unit 1 along the x, y, and z-axis. Respectively, these are called i, j, and k, like this photo shows:
You can convert any vector into a unit vector though:
All you have to do is take each component (x, y, z) of the vector, and divide it by the magnitude of the vector (using the equation above to find the magnitude first):
