>> In this section, we're going to begin a study of vectors, and to get an idea of what vectors are all about, consider this problem. Suppose we have a ship that is heading east at 15 miles per hour. Now, heading refers to the direction that the bow of the ship is facing, not necessarily the, the path which it is traveling, as we'll see in a moment. Because there is a tide affecting the travel of the ship, you see, and the tide is traveling north ten degrees east at a rate of five miles per hour. We want to find the speed and the course of the ship. Although the ship is facing east, it's not actually traveling east because of the effect of the tide. Now this is the notion of, of the idea of vector, and both the ship and the tide represent a vector idea, because they have involved with them both a speed and a direction. So vector quantities are quantities that have direction and magnitude associated with them. It requires, then, two real numbers to describe a vector quantity. On the other hand, scalar quantities have magnitude only associated with them and can be described using one real number. Well, to, to solve the problem, we can bring to bear the, many of the techniques that we have learned already involving trigonometry, and we're going to take a geometric approach to this problem, and in fact we're going to see a couple of geometric approaches. And then we're going to move into an algebraic approach to the same kind of problem a little bit later. Well, the geometric approach looks like this, or one of the geometric approaches looks like this. We can represent the ship vector and the tide vector using a directed segment. Now for the ship, it's a, a, a segment which has direction given by the arrow at the end of the segment. Now there's a, an initial point A and a terminal point B involved with this vector idea, with this little arrow arrangement. The direction of the arrow corresponds with the direction of the ship. Remember, the, the ship is going due east, so we, we make our vector arrow go, we cause it to go due east you see. And its length corresponds with the magnitude associated with the ship. The ship is going 15 miles per hour, so we have a, a certain distance associated with that. Now, if the, since the tide is going in a different direction at a different speed, you see, we have a different arrow direction for its vector, and we have a different length associated with it because it is less in, the speed is a little bit less. The magnitude is a bit less. Well, let's see. We have the, the ship is going 15 miles per hour, so this rascal would be 15. The tide is going five miles per hour, so we can associate it with a number five. Now, the, the ship is, is designated, the notation for the ship vector is AB. The A is written first because it's the initial side, the B is written second because it's the terminal side, and this little notation simply indicates that it's a vector, and it starts at A and goes to B. So, so the direction is built into this notation. The tide vector is AC, and again, same idea.