Vectors Vector Practice Problems: Odd-numbered problems from 3.1-3.21 Reminder: Scalars and Vectors Vector: Scalar: A number (magnitude) with a direction. Just a number. I have continually asked you, which way are the v and a vectors pointing? av Vectors A car drives 50 east and north. What is the displacement of the car from its starting point? 50 Vectors A car drives 50 east and north. What is the displacement of the car from its starting point? Displacement is a vector (net change in position) 50
Describing a vector A vector is described *completely* by two quantities: magnitude (How long is the arrow?) & direction (What direction is the arrow pointing?) Magnitude and direction Magnitude: length of this line θ 50 Direction: angle from reference point (here, θ degrees of ) Vector notation c d This vector written down: A A cd And its magnitude A A cd Vector components Ay Ay Vector change in y direction Ax Ax Vector change in x direction
Translating vectors Vectors are defined by ONLY magnitude and direction. = = = These are all the SAME vector! Multiplying by -1 V, has an equal magnitude but opposite direction to V. = In which case does =? A. B. C. D. Q17 Geometrically adding vectors [m/s] [m/s] Two vectors with the SAME UNITS can be added.
Geometrically adding vectors + =? tail tip Geometrically adding vectors + =? tail tip When adding geometrically, always add tail to tip! Geometrically adding vectors + = vector + vector = vector This is called the triangle method of addition Geometrically adding vectors + = + = It s commutative! It doesn t matter which one you add first.
If you were to add these two vectors, roughly what direction would your result point? Q18 A. B. C. D. E. None of the above If you were to add these two vectors, roughly what direction would your result point? Q18 A. B. C. D. E. None of the above If you were to add these two vectors, roughly what direction would your result point? Q18 A. B. C. D. E. None of the above V1 + V2 = VR Translate the vector and always add tail to tip!
What is + =? Q19 A. B. C. D. E. None of the above Geometrically subtracting vectors - =? When adding/subtracting geometrically, always add tail to tip! Geometrically subtracting vectors - - = + (- ) When adding/subtracting geometrically, always add tail to tip! Geometrically subtracting vectors - - = + (- ) When adding/subtracting geometrically, always add tail to tip!
Geometrically subtracting vectors - - = + (- ) When adding/subtracting geometrically, always add tail to tip! Geometrically subtracting vectors - - = + (- ) When adding/subtracting geometrically, always add tail to tip! Vectors A car drives 50 east and north. What is the displacement of the car from its starting point? 50 Vectors A car drives 50 east and north, then 20 south. What is the displacement of the car from its starting point? 20 South
R Fig. 3.4 in your book In graphical addition/subtraction, the arrows should always follow on from one another, and the resultant vector should always go from the starting point to the destination point in your summed vector path. Scalar multiplication 3-3 Multiplying a vector A and a scalar (i.e. number) k makes a vector, denoted by ka. Intermission Vector arithmetic: components 50 Dx Dy What is D (the magnitude of )? A. 58 B. 80 C. 20 D. 0 E. 58 m/s Q20
Vector arithmetic: components 50 Dx Dy What is D (the magnitude of )? A. 58 B. 80 C. 20 D. 0 E. 58 m/s Q20 Think about the Pythagorean Theorem Vector arithmetic: components A Ay Ax A, Ay, and Ax here are the MAGNITUDES of the vectors drawn (they don t have hats and are not bold). Vector arithmetic: components A Ay Ax The magnitude of a vector component is its final number minus initial number! Vector arithmetic: components yf A Ay Ay = Ay = yf - yi xi Ax yi xf Ax = Ax = xf - xi
Vector arithmetic: components What is the magnitude of the x component of D? D 70 o Dx Dy A. 60.6 B. 35.0 C. 40.4 D. 0 E. 31 degrees Q21 Vector arithmetic: components What is the magnitude of the x component of D? D 70 o Dx Dy A. 60.6 B. 35.0 C. 40.4 D. 0 E. 31 degrees Q21 Think about SOH CAH TOA! Using trigonometry, you can find all vector components and angles given just a bit of information! Look at the triangles, and think about what you can figure out based on available info. Ultimate rule of vector math Don t fear the vector. To study: Practice drawing/graphing vector operations. Get used to vector and magnitude notations. Practice solving for x, y components. Practice solving for θ.
Diandra kicks a soccer ball to a max height of 5.4 m at a 20 angle from the ground with a speed of m/s. What is the x (horizontal) component of the initial velocity of the soccer ball? Diandra kicks a soccer ball to a max height of 5.4 m at a 20 angle from the ground with a speed of m/s. What is the x (horizontal) component of the initial velocity of the soccer ball? TO START: Draw your vector right triangle. What are the sides? Compare your triangle with your neighbor. CAREFUL! You can t do vector arithmetic combining displacement (5.4m) with speed (m/s)! 20 m/s 5.4 m