Introduction to Vectors and Their Operations

Transcript

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.