Vectors. Chapter 3. Arithmetic. Resultant. Drawing Vectors. Sometimes objects have two velocities! Sometimes direction matters!

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1 Vectors Chapter 3 Vector and Vector Addition Sometimes direction matters! (vector) Force Velocity Momentum Sometimes it doesn t! (scalar) Mass Speed Time Arithmetic Arithmetic works for scalars. 2 apples + 3 apples = 5 apples Arithmetic doesn t work for vectors. 2 N + 3 N may or may not = 5 N (it depends on their directions!) Sometimes objects have two velocities! Boats Velocity from the motor Velocity from the current Airplanes Velocity from the engines Velocity from the wind Resultant When two numbers are added together, the answer is called the SUM. When two vectors are added together, the answer is called the RESULTANT. The resultant has two parts and both must be given in the answer. Speed Direction a scale is clearly listed a vector arrow (with arrowhead) is drawn in a specified direction. the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North). Drawing Vectors 1

2 Conventions for Direction Headings If all vectors were N,S,E,W it would be easy. But what if they are NW or NNE? We a better system. Arrows are Used to Represent Vectors Arrows can be long or short to show magnitude. Arrows can point in the direction of the force or velocity. Try these Direction and Magnitude of Vectors Adding Vectors A boat, whose engine can move it through the water at 10.0 m/s, heads east across a river that flows south at 4.00 m/s. How fast and in which direction will the boat move? 1) Sketch the vectors head to tail. 2) Sketch the resultant (answer). 3) Determine the magnitude (size) of the resultant. 4) Determine the direction of the resultant. 5) Report size and direction of the resultant. 2

3 1. Sketch the vectors head to tail on a grid (sketch in the coordinate axis. 2. Sketch the resultant. It goes from the tail of the first vector to the head of the second. Determine the magnitude of the resultant using the Pythagorean Theorem. (python what?) Try these and text answer In each case, use the Pythagorean theorem to determine the magnitude of the vector sum. Determine the heading of the resultant. (SOH CAH TOA) SOH CAH TOA 3

4 Report your answer with magnitude and direction. Try these and text your answer Test your understanding of the use of SOH CAH TOA to determine the vector direction by trying the following two practice problems. In each case, use SOH CAH TOA to determine the direction of the resultant What if there s more than 2 vectors to add? A student drives his car 6.0 km, North before making a right hand turn and driving 6.0 km to the East. Finally, the student makes a left hand turn and travels another 2.0 km to the north. What is the magnitude of the overall displacement of the student? Visualize the vectors before adding them Add in head to tail method the resultant is independent by the order in which Vectors are added Same thing Switch the order around of the vecotrs to make a right triangle. Now the PT can be used As can be seen in the diagram, the resultant vector (drawn in black) is not the hypotenuse of any right triangle, so Pythagorean Theorem can t be used. Or can it? What is the magnitude of the resultant? 4

5 The direction of the resultant? Observe that the angle in the lower left of the triangle has been labeled as theta (Θ). Theta (Θ) represents the angle that the vector makes with the north axis. Theta (Θ) can be calculated using one of the three trigonometric functions introduced earlier in this lesson - sine, cosine or tangent. The mnemonic SOH CAH TOA is a helpful way of remembering which function to use. TOA Tangent(Θ) = Opposite/Adjacent Tangent(Θ) = 6.0/8.0 Tangent(Θ) = 0.75 Θ = tan -1 (0.75) Θ = Θ =37 But you re not done The problem is not over once the value of theta (Θ) has been calculated. This angle measure must now be used to state the direction. One means of doing so is to simply state that the direction of the resultant is 37 east of north. Alternatively, the counter-clockwise convention could be used. Since the angle that the resultant makes with east is the complement of the angle that it makes with north, we could express the direction as 53 CCW. If you don t like trig Draw the right angle additions instead! Objective: Resolve a vector into horizontal and vertical components. Components--two vectors at right angles to each other that are derived from a single vector. Resolution--the process of determining the components of a vector. Vertical Compone nt of Velocity Horizontal Component of Velocity Velocity of Ball Vector Resolution: Steps First, draw vertical and horizontal lines from the tail of the vector. Y V Second, a rectangle is drawn that encloses the vector as its diagonal. The sides of the rectangle are the components. Y X V X 5