It’s Professor Dave, I wanna tell you about vectors.
In math, we are always dealing with numbers.
Big, small, rational, irrational, but numbers all the same.
Now it’s time to learn about another concept called a vector.
This is a mathematical construct that has both magnitude and direction.
The magnitude will be a number, so a vector can have a magnitude of five.
This is the part that tells us how much.
But it also has a direction, meaning five of something in a particular direction.
This is why we represent a vector with an arrow.
The length of the vector tells us the magnitude, so a longer vector means a greater magnitude,
and the direction the arrow points, relative to some axes, tells us the direction of the
vector.
Vectors are of tremendous importance in physics, because they describe things like forces that
cause the three-dimensional motion of objects.
You can check out some of the things we can do with vectors after this, by checking out
my classical physics series, but for now, let’s go ahead and learn all the things
we need to know about vectors in order to do more math.
Many vectors are simply displacement vectors, denoting how far an object has traveled and
in which direction.
If such a vector connects two points, like A and B, we can call this vector AB with a
little arrow over it.
That’s one way to represent a vector.
We can also just refer to a vector with a letter, like U.
Different vectors with the same magnitude and direction are called equivalent vectors.
But of course, vectors can have different magnitudes and directions, and we can manipulate
and combine vectors in various ways.
First there is vector addition.
If we have vector U and vector V, and we place them head to tail, with V starting where U
ends, being very careful not to alter the direction of either vector, then the sum of
these vectors, or U plus V, will simply go from the start of U to the end of V, forming
a triangle, like so.
If these vectors were AB and BC, their sum would be called AC, as it would connect A
and C.
We don’t necessarily have to draw triangles, though.
Let’s say U and V start at the same point.
Let’s duplicate U and V and move them so as to form a parallelogram, like this.
Now the sum of the two vectors, U plus V, can be represented as the diagonal of the
parallelogram.
So there are a number of ways to represent vector addition geometrically.
We can also perform multiplication with vectors.
The simplest way to do this is called scalar multiplication.
This is when a vector is multiplied by a scalar, which is just a number.
So if we have some vector U, and we multiply it by the scalar, two, all we do is double
the length of the vector, making it start from the same point, but reach twice is far
in the direction it is pointing.
So when multiplying a vector by a scalar that is greater than zero, we just multiply the
length of the vector by the scalar.
If the scalar is less than zero, or a negative number, we still alter the length of the vector
in the same way, but we also reorient it to point in the negative direction relative to
its original position, which is just precisely the opposite direction, or 180 degrees away.
So vector U times the scalar negative one, will just flip it over in this manner, to
give us negative U.
This understanding will allow us to now perform vector subtraction.
If we have two vectors, U and V, aligned as we did previously, how can we get U minus V?
Well U minus V is the same thing as U plus negative V.
So if we take V and flip the direction, it is now negative V, and we can just do vector
addition like we already learned.
Completing the triangle with U and negative V will give us U plus negative V, or U minus
V, and we have just performed vector subtraction.
We can combine scalar multiplication and vector addition and subtraction to do all kinds of
things with vectors, which we will see later.
We should also note that vectors can be represented by components, which are kind of like coordinates.
Let’s say a vector extends from the origin to the point A one, A two.
The vector itself can be represented this way, kind of like the point, but with this
bracket-like notation rather than parentheses, to distinguish the vector from the ordered
pair that represents the point.
This can be done in a three-dimensional coordinate system as well, with X, Y, and Z axes.
The length or magnitude of a vector, represented by these brackets here that look like absolute
value brackets, can be calculated by finding the square root of the sum of the squares
of its components.
We can also use this information to do more algebra with vectors.
If we have a vector A with components A one and A two, and another vector B with components
B one and B two, the sum of these vectors, or A plus B, will have the components A one
plus B one and A two plus B two.
The same can be said for vector subtraction, we just subtract the corresponding components.
In fact, let’s quickly mention some important properties of vectors.
The commutative and associative properties of addition apply to vectors, like the way
that A plus B equals B plus A, and the way that A plus the quantity B plus C equals the
quantity A plus B, plus C. Any vector plus the zero vector, which is the vector with
zeros for all its components, will simply give the same vector we started with.
Any vector plus the negative version of that same vector, which is simply a vector minus
itself, is the zero vector.
While we can’t add a vector and scalar, a scalar can be distributed across a sum of
vectors, and a vector can be distributed across a sum of scalars.
A product of scalars times a vector is the same as one of the scalars times the product
of the other scalar and the vector.
And one times any vector is simply that vector.
Those are the important properties we should understand.
There is one other way we can represent vectors that will be important to know, and it involves
standard basis vectors, which are unit vectors, meaning they have a length of one.
If we are in a three-dimensional coordinate system, any any vector can be represented
by three components.
The vector with the components one, zero, zero, extending one unit along the X-axis,
is denoted by a lower case I.
The vector with components zero, one, zero, extending one unit along the Y-axis, is denoted
by a lower case J. And zero, zero, one, extending one unit along the Z-axis, is denoted by a
lower case K.
We can now represent any vector with multiples of I, J, and K.
So five, negative two, nine, can also be five I minus two J plus nine K.
This also works in two dimensions, we just lose the K.
The reason this is useful is that algebraic manipulations can become much more intuitive,
as operations like multiplying a vector by a scalar are reduced to simply distributing
a number across a binomial or trinomial, which is trivial algebra.
We can also add two vectors together without having to draw them, we can simply define
one vector, A, which we can say is equal to two I minus J plus four K, and another vector,
B, which is I plus six J minus two K.
Then to do something like two A plus three B, we can simply change A and B to this other
notation.
First we distribute each scalar across the sum it operates on.
Then we simply combine like terms, and we end up with seven I plus sixteen J plus two
K, and we have performed this complex operation using simple algebra.
Now that we know how to perform a few different operations with vectors, let’s check comprehension.