Let’s start with the basics
Vectors are mathematical tools for describing things that have both size and direction.
Imagine you’re describing how to get somewhere. You wouldn’t just say “go 25 kilometers,” you’d also say which way to go, right?
That’s what vectors do! They’re mathematical objects that capture both direction and magnitude. So, if a bike travels 25km west from point A to point B, we can neatly represent that information as a vector.
The vector captures the bike’s displacement. Its magnitude reveals the distance traveled (15km), and its direction specifies the direction (west).
The vector starts at point B (the tail) and ends at point A (the head). We can represent this vector using its endpoints, writing it as
Let’s say we need to illustrate the displacement vector of a bike that travels 15km on a bearing of 60 degrees. What would that diagram look like?
Let’s pick a starting point for our vector anywhere on the plane. We’ll use a vertical line to show which way is north.
we go 60 degree clockwise from the north
equal vectors !
For vectors to be considered equal, they must have the same length and point in the same direction. The example vectors in the image illustrate this concept.
From the picture above, we can say
Understanding how to add vectors using the triangle law.
To add two vectors together, a visual method is to connect them head-to-tail. to be more descriptive “The sum of two vectors can be found graphically by positioning the starting point (tail) of one vector at the ending point (head) of the other.”
Adding ‘a’ and ‘b’ involves a simple geometric construction. First, place the tail of vector ‘b’ at the head of vector ‘a’. Then, the vector ‘a+b’ is the one that connects the tail of ‘a’ to the head of ‘b’.
This method of adding vectors head-to-tail is known as the triangle law, and the resulting vector from this addition is often called the “resultant” vector.
opposite and the zero vector
Imagine a vector going from point A to point B (AB). The opposite vector (BA) travels the same distance but goes from point B back to point A.
we write
To compute the sum of two opposite vectors,
we can use the triangle law of addition.
The triangle law of addition tells us that when adding vectors, the result (“resultant”) is a vector drawn from the starting point of the first vector to the ending point of the last. If we end up back where we started (the tail and head meet),
we get a special vector called the zero vector, denoted as 0. It has no direction and zero length, meaning its head and tail are at the same location.
Understanding how to subtract vectors using the triangle law.
We’ve learned how the triangle law helps us add vectors. But did you know we can also use it for subtraction?
The secret is to realize that subtracting a vector is the same as adding its inverse. So, the operation a – b is identical to a + (-b). Remember, -b is simply b pointing the other way, with the same length. Let’s clarify this with an example.
Subtracting vector b from vector a (written as a – b) can be thought of as adding a to the opposite of b (written as a + (-b)). The vector -b has the same length as b but points in the exact opposite direction.
To visualize this, imagine drawing vector a anywhere on a plane. Then, starting from the tip (head) of vector a, draw vector -b. Now, the resulting vector (a – b) is the one that connects the starting point (tail) of a to the ending point (head) of -b. This follows the triangle law of addition, but with -b instead of b.”
Figuring out the length or size of a vector.
Let’s say we have a vector that goes from point A to point B, and we call it “a”. The size or length of this vector – also known as its magnitude or modulus – is written as |AB| or |a|.
For instance, if a vector (let’s call it AB) has a length of 8, we’d write that as |AB| = 8.
Now, imagine this: vector “a” points straight up and has a length of 5 (|a|=5). Vector “b” points directly to the right and has a length of 8 (|b|=8). How would we find the length of the vector we get when we add “a” and “b” together (i.e., what is |a+b|)?
The first step is to use the triangle law of vector addition to figure out what the resulting vector (a+b) looks like.
Looking at the diagram, we can see a right triangle is formed. This allows us to use the Pythagorean theorem to find relationships between the sides.
= 25 + 64
= 89
therefore, |a+b|= √ 89 = 9.43
Important: We just used the Pythagorean theorem, but be careful! This only holds true when the vectors ‘a’ and ‘b’ are perpendicular (at right angles). This isn’t always the case! Next, we’ll tackle how to work with vectors that are not at right angles.
Calculating the magnitude of a vector using the law of cosines
if adding vectors using the triangle law doesn’t result in a right triangle, we can’t rely on the Pythagorean theorem to easily calculate the magnitude of the resulting vector.
For instance, when we add vectors ‘a’ and ‘b’ and the resulting triangle isn’t a right triangle (instead, it has an angle greater than 90 degrees, called an obtuse angle, theta), the Pythagorean theorem won’t work directly.
how ever we can use the law of cosines.
If you have two vectors and you visualize their sum as forming a triangle, you can always calculate the length of the sum vector using the law of cosines.