### Vector Presentation

Transcript: Named after it's discoverer, Pythagoras, it is one of the earliest theorems known to civilization. It is a part of Euclidean geometry and deals with only right triangles. The definition of the Pythagorean theorem is "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." Basically, a^2 +b^2 = c^2 Cosine The cosine function is used to find the x component of the vetor. Cos = adj/hyp adj is the adjacent side of the angle . hyp is the hypotenuse. As with the sine function we will have to rearrange the equation to find the adjacent side (x componet). The equation will look like: (hyp)Cos = adj Sine The sine function is used to find the y component of the vector. Sin = opp/hyp opp is the opposite side of the angle . hyp is the hypotenuse side of the triangle. When you are given a vector you are given the angle and the hypotenuse. You need to find the opposite side (y component). In order to do this we will have to rearrange the equation like so: (hyp)Sin = opp 35º Adding Non-Perpendicular Vectors One of the easiest and quickest ways to define a vector is to use the Pythagorean theorem and the tangent function. After you use the tail to head method to graph your vectors you connect the tail of the first to the head of the last. Now, using the Pythagorean theorem, you square your adjacent and opposite sides, add them together, find the square root of the sum, and you have the magnitude of the resultant. Next, you use the tangent function and you have the heading of the resultant. Remember that tan=opp/adj. Unfortunately none of this will work if the vectors do not form a right triangle. Vector Operations Example Given: v=95 km/h =35º Unknown: vx=? vy=? Rearranged equations: vy = vsin vx = vcos vy = (95 km/h)(sin35º) vy = 54km/h vx = (95km/h)(cos35º) vx = 78km/h What happens when you are given a resultant vector and you need to fingure out its components? In order to find the components you have to resolve the vector. Resolving the vector breaks the vector down into its x and y components. But how do you do this? Vectors When adding non-perpendicular vectors you must break each vector into its components. What this means is that you must break each vector into its x and y components. After you have broken each vector into its component you can add their x and y values together. Plot the resulting x and y coordinates and now you can use the Pythagorean theorem and tangent function to find the resultant. vy Tangent is defined as opposite/adjacent. It is very useful because you can enter the side lengths of two sides and find the degree of any angle, depending on which two side lengths you enter into the equation. When making a coordinate system you plot two different values along the x and y axis respectivly. A common coordinate system is a compass. It uses the y axis as north and south and the x axis as west to east. v = 95km/h vx Resolving Vectors Tangent Function There are two equations used to find the components. Sin = opp/hyp Cos = adj/hyp These are the trig funtctions sine and cosine. Using the theorem and tangent The Pythagorean Theorem Coordinate systems In order to solve a problem involving coordinate systems you have to use one that can actually solve the problem. For example if you were trying to measure the velocity of a car you couldn't use a coordinate system that measured altitude. Using sine and cosine To find the x component of a vector you need to use cos. You multiply the magnitude of the vector by the cos of the degree of the vector. Finding the y component is just as easy, you just swap sin in for cos. Abbreviated sin and cos