 Physical properties like area, volume and temperature are scalars since we only need a number to represent them. But when we deal with other properties such as velocity and acceleration, we may use both the size and the direction. Such objects which have a direction and a size or magnitude are called vectors. For example, if we move an object in some direction with a force, we describe that force as a vector or if we're moving with constant velocity. Assume that we are moving from point A to point B and our velocity is 5. It means that our vector has a magnitude of 5 and an AB direction. We denote the vector from A to B as AB. Or we can just name it with one of the lower case letters, but also including an arrow or a line on the top. For example, A. We call B a head and A a tail. The length of that segment is our vector's magnitude. You may say, hey, but it's just a line segment with an arrow and you would be almost right. The difference is that we can move the vector and it would remain unchanged, but not the line segment. These three vectors, for example, are identical because they have equal magnitude and the same direction. So what can we do with vectors? Let's put this one on a coordinate plane. We can put it anywhere, but for now let's move it so its tail has coordinates 00. If the coordinates of the tail A is 00, then the coordinates of the vector AB are coordinates of its head B. Easy. Our vector is now just two numbers, the coordinates of the head. Assume that B has the coordinates 43. Then AB equals the square root of the sum of 4 squared and 3 squared, which equals 5. And that we know because of the Pythagoras theorem. Generally speaking, if vector AB has coordinates XY, then its magnitude can be found using this formula. We can even add vectors. Imagine that you are pulled by two forces F1 and F2 at the same time. Which direction will you move? Let's force ourselves step by step. Let's allow F1 to do it first and then F2. We do that by moving F2 in such a way that its tail becomes attached to the head of F1. Then we connect the tail of F1 and the head of F2. The vector formed is the sum of F1 and F2. Another method would be to put the tails of the two vectors together and draw a parallelogram like this. The diagonal vector that starts from the tails of F1 and F2 will be the sum of these vectors. We can also just add corresponding coordinates of two vectors and get the coordinates of the sum vector. For example, if we have vectors A with coordinates 53 and B with coordinates 24, then C equals A plus B. And so we have the coordinates of vector C, which is 7, 7. We can also multiply a vector by a number. If the number is positive, the direction will remain the same and the magnitude will be multiplied by this number. If the number is negative, we change the direction of the vector to opposite and then multiply the magnitude. Or just multiply both coordinates by that number. If vector A has coordinates XY, then vector N times A will have coordinates NX, NY. To find vector A minus vector B, just multiply vector B by minus 1, which changes its direction to opposite and add. You already know how to add vectors. And now let's use this new knowledge. There are two vectors, vector A with coordinates minus 3 and minus 1, and vector B with coordinates 1 and minus 2. Find this vector and its magnitude. We can analyze it even without drawing, as it's one of the features of vectors we already know. First, we find the coordinates of vector minus 2B, then the coordinates of vector A minus 2B. And finally, the magnitude of this vector. Pretty easy, isn't it? We simply completed the same operation with each of the coordinates. Now let's look at what happened on the coordinate plane. First, we found vector minus 2B. Then, we added vector A and vector minus 2B. After that, we use the Pythagoras theorem to find the length of that vector. Can you find the coordinates and the magnitude of this vector? Vectors A and B have the same coordinates as in the previous task. Pause the video and give it a go. Vector minus 2A plus 3B has coordinates 9 and minus 4. And its length is the square root of 97. Did you get it right? If you liked the video, give it a thumbs up. And don't forget to subscribe. Comment below if you have any questions. Why not check out our FuSchool app as well? Until next time.