Professor Dave here, let's talk about scalars and vectors. Physics is all about some pretty amazing stuff, from subatomic particles to everything you see in space. But before we dive into all that, just as with any other science, we have to get a few things out of the way in terms of language and convention. To learn about science we have to know how people do science, and how science is communicated, so we must first know about things like the units of measurement that the scientific community uses, scientific notation which is used to report very small and very large numbers that will be common in physics, and dimensional analysis for converting between units. Luckily we've already gone over these in my chemistry series, but in case you missed those tutorials you may want to check them out at some point. Beyond these basics, one other prerequisite for studying physics is the understanding of scalars and vectors. To put it briefly, a scalar is a quantity that communicates a particular size or magnitude. These can be things like the mass of a book, the amount of time you spend watching this clip, or the temperature outside today. These are numerical quantities along with any relevant units that simply answer the question: How much? A vector is different because a vector has not only magnitude but also direction, so it answers both: How much? And: Which way? Vectors will be very important in physics because when we look at forces we want to know how much force is being applied but also in which direction. These forces will always be represented by vectors where the direction the arrow is pointing is important information just as much as the length of the arrow which will indicate the magnitude of the vector. To denote vectors we usually use letters in bold font with an arrow on top. As we learn more about physics we will use vectors to predict the motion of objects, and if there are several forces acting on an object, as there often are, we will need to be able to do certain operations with vectors in order to perform our calculations, so let's learn how to do simple vector operations. If we want to add two vectors together it's pretty simple, we just make the second vector start where the first vector ends and then create a new resultant vector that goes from the start of the first to the end of the second. In lining up two vectors we have to make sure they retain their direction, as that is important information. If the vectors point in the same direction their sum will just be a longer vector, and the magnitude of this resultant vector will simply be the sum of the magnitudes of the original vectors, so in this case 4 plus 3 equals 7. Sometimes there is an angle between the two vectors, like these perpendicular vectors. In this case, when we draw the resultant vector it will be the hypotenuse of a right triangle. The magnitude of this resultant vector is not the sum of the other two magnitudes, this one can be found by using the Pythagorean theorem, which we may remember from algebra class. A squared plus B squared equals C squared. Plug in A and B, square them and find the sum, and then we take the square root to get C, which in this case will be 5. When we are looking at right triangles we can also use trigonometric functions to find the unknown angles, all we need to do is choose an angle which we represent with the Greek letter theta, and label the sides of the triangle as opposite, the leg directly opposite the angle, adjacent, the leg right next to the angle, and the hypotenuse, which is always the longest side. We can use the lengths of these sides to find the sine, cosine, and tangent of this angle. We will cover these functions in more depth in the upcoming mathematics course, but for now all we need to remember is the mnemonic device: SOHCAHTOA. Sine theta is equal to opposite over hypotenuse, in this case that would be 3/5. Cosine theta is equal to adjacent over hypotenuse, which here will be 4/5, and tangent theta is opposite over adjacent, or 3/4. To solve for theta we can use any of these equations. Using this one we can take the inverse sine of both sides, that makes theta equal to the inverse sine of 3/5. Just put 3/5 or 0.6 into your calculator and press the inverse sine button which may look like any of these expressions, and we get about 36.9 degrees. Trigonometric functions like these relate the angles of a right triangle to the length of the sides and we will use them a lot in physics, but it usually won't be more complicated than this so don't worry too much about it. Also, if you are using a graphing calculator take a moment to familiarize yourself with the different settings it uses to report angles, which are degrees and radians. If you are more familiar with degrees make sure that setting is selected and we will learn about radians another day. Now that we understand vector addition let's quickly go through a few other operations. Vector subtraction will be similar to vector addition except that when subtracting vectors we will again line them up head to tail but we will invert the direction of the second vector. This is the same as multiplying the vector by a scalar of negative 1. This way the magnitude of the vector remains the same but its direction is reversed, so instead of A plus B which looks like this, A minus B would look like this, which is essentially A plus negative B. We find the resultant vector and that's all there is to it. Another thing we might want to do is break up a vector into X and Y components. This is useful when looking at projectile motion. If a ball is thrown at an angle, its velocity can be represented with a vector, but because gravity is operating on the ball in the Y-direction pulling it down to the earth, we will want to be able to separate its vertical motion from its horizontal motion, so we can take this vector and split it up into components. All we need to know is the magnitude of the vector and its angle from the horizontal, then we can use trigonometry to find the length of the legs of a hypothetical right triangle. Sine of theta gives us opposite over hypotenuse and cosine theta gives us adjacent over hypotenuse. On both of these we can evaluate the trig function and multiply both sides by the hypotenuse, which is the value we know, to get the legs of the triangle which will be the X and Y components of the vector. We will see this in more detail later. Vectors can also be multiplied by scalars or other vectors. When multiplying a vector by a scalar we will simply multiply the magnitude of that vector by the scalar while keeping the direction the same, so if multiplying this vector by 2, its length will double. Two vectors can also be multiplied but that will be more complicated, so we will go over that another time. Let's try one example to make sure we can do these kinds of calculations. Say you are in a boat traveling across a river and the boat typically moves at 4 meters per second in still water. But the current is pushing you downstream at a rate of one meter per second. What is the speed and direction of the boat? To answer this and almost any other problem we study in physics it is best to draw a diagram first. Here's a top-down view of the river and the boat. Let's draw a vector that is horizontal with a magnitude of 4. We then need another vector to represent the current. That will point straight down and have a magnitude of 1. To find out the speed and direction of the boat we have to add these vectors together. In order to do this we can move the vector that represents the current over here so that we are adding head to tail and then find the resultant vector, or we could have left it where it was, made a rectangle using the two vectors and then just drawn the diagonal. Either way this vector sum will have a magnitude of root 17. To find the angle from the horizontal we can use any trig function we like. Say we use tangent. tangent theta will be opposite over adjacent, or one-fourth. Taking the inverse tangent, theta must be 14 degrees, so the boat is moving at a speed of root 17 meters per second in the direction of 14 degrees off the horizontal. It may seem abstract right now but we will be using this sort of approach in order to understand motion in almost any situation you can imagine because it's just about using vectors to represent the forces acting on any object. Now we are ready to move on to the physics, but first let's check comprehension. Thanks for watching, guys. Subscribe to my channel for more tutorials, support me on patreon so I can keep making content, and as always. feel free to email me: