In order to do more complex calculations with vectors, we need to understand their algebraic representation. While the geometric representation of vectors makes it easier for us to understand them, the algebraic representation makes it easier for us to perform simple calculations quickly such as addition and scalar multiplication. The algebraic representation uses the numerical components of the vector.
I want to talk about the Algebraic formula of vector. Let's start with the concept of position vector. Let p be a point in the xy coordinate system, here is p, vector op is the position vector of point p where o is the origin and p is p you're drawing a vector from the origin to point p. This is the position vector of point p. Now if x and y are the coordinates of point p, x is the horizontal component of vector op and y is the vertical component and that means we can denote vector op in component form this way x comma y. Now the length of magnitude of a vector in component form if you have a vector in component form, then its magnitude is the square root of the sum of the squares of the components so for example here the magnitude would be the square root of x squared plus y squared. Now let's do an example, let's sketch each of these vectors and find its length. Okay, let's start with 3, 4. 3, 4 is going to go from the origin point o to the point 3,4 and that's here and so this is going to be your vector your position vector for the point. Now, what's the length? Well you take the square root of 3 squared plus 4 squared and you get root 9 plus 16 root 25 and that's 5. What about this one? -4, 2 let's draw out in another color, that's going to go from 0 to 1, 2, 3 -4 1, 2 here, so that's the vector -4, 2 and its length, the square root of -4 squared plus 2 squared that's 16 plus 4 root 20 and root 20 is simplifies to 2 root 5 so that would be the length of this vector. Okay, let's do another. -5, -5 that's the vector from the origin to the point 1, 2, 3, 4, 5 -5, 1, 2, 3, 4, 5 -5 it's right there and so I draw my vector, and the length of that vector is the square root of -5 squared plus -5 squared that's 25+25 root 50 and root 50 is the same as 5 root 2. Finally 0, -8 I'll go back to board again 1, 2, 3, 4, 5, 6, 7, 8 is down here 0, -8 is this vector that points straight down and you can actually see that its length is 8 so you don't need to use the the formula here so the length of this vector is 8. Now again this is the component form for a vector and when you're graphing this vector, you draw a a vector from the origin to this point 3, 4 and to find the length of such a vector you just take the square root of the sum of the squares of the two components.