Main Ideas in Class Today

Transcription

1 Main Ideas in Class Today After today, you should be able to: Understand vector notation Use basic trigonometry in order to find the x and y components of a vector (only right triangles) Add and subtract vectors Practice Problems: 3.1, 3.3, 3.5, 3.7, 3.9, 3.11, 3.13, 3.15, 3.17, 3.19, 3.21

2 Quick Review Quantities that are determined by a magnitude alone are called scalars. Quantities that have both magnitude and direction are called vectors. e.g. displacement, velocity, acceleration

3 A vector may be represented geometrically by a directed line segment or an arrow whose direction represents the direction of the vector and whose length represents the magnitude. The vector may be called by AB, a, and its magnitude is denoted by AB, a, Vector Notation Varies a, a, a a x f = 50 m B A x 0 = 0 m

4 Two vectors are said to be equal if and only if they have the same magnitude and direction. ABCD is a parallelogram, A D then AB DC and AD BC B C How about AB and CD?

5 The negative vector of v, denoted by v, is a vector having equal magnitude but opposite direction to v. Therefore AB BA A A v -v B B

6 Basic Operation of Vectors: Addition C triangle law of addition AB BC AC A Vector tail B Vector tip

7 Basic Operation of Vectors: Addition triangle law of addition A Vector tail B Vector tip

8 Basic Operation of Vectors: Addition commutative law of addition 1+2=2+1 C AB BC AC BC AB AC A B

9 Basic Operation of Vectors parallelogram law of addition OA OC OB C B O A

10 Addition: Vector Arithmetic V V V 1 2 Result triangle law of addition It doesn t matter which order you add; the answer is the same.

11 Recall: Negative of a vector: a vector having the same length (magnitude) but opposite direction Subtraction Which direction should a-b point? b B a b = a + (-b) O a A -b a + (-b)

12 Scalar Multiplication The product of a vector a and a scalar k is a vector, denoted by ka. This operation is called scalar multiplication. If k = 3. a 3a -3a If k is negative, then reverses direction.

13 Vectors in the Rectangular Coordinates System A rectangle is a special case of a parallelogram where angles equal 90. OMPN is a rectangle, OP OM ON How to find the magnitude of OP?

14 y-component Vector Arithmetic Components Components of a vector (commonly velocity) Recall for right triangles: x-component a 2 b 2 c 2 Pythagorean Theorem 2 2 v v v 2 x y

15 Vector Arithmetic Components Components of a vector (commonly velocity) Recall for right triangles: These are formulas with three variables. If you know 2, you can solve for the other. sin cos tan opposite hypotenuse adjacent hypotenuse opposite adjacent adjacent (adj) opposite (opp)

16 Only true if angle adjacent to x axis! Otherwise, go back to definitions. sin cos tan opposite hypotenuse adjacent hypotenuse opposite adjacent V V x y V V cos sin (Useful if V and are known) These switch if angle defined from y axis. V V 2 x V tan V V y x 2 y (Useful if components are known) V x and V y switch if angle defined from y axis.

17 Diandra kicks a soccer ball at a 20 angle from the ground with a speed of 30 m/s to a max height of 5.4 m. What is the x (horizontal) component of the initial velocity of the soccer ball? Hint: Draw your vector right triangle. What are the sides? Compare your triangle with your neighbor.

18 Common problem: Make sure your triangle sides all have the same units! m/s X 5.4 m

19 Vector Arithmetic Components (Will be important in Chapter 4) Addition: V V 1 V2 Vx V1 x V2 x Vy V1 y V2 y When adding vectors, components are added separately Never add magnitudes of vectors

20 The total amount that you go East, is the amount you go East on Day 1 plus the amount that you go East on Day 2. What would I do if I backtracked some? A hiker goes on a 2-day hike. On the first day, the hiker travels 25 km Northeast. On day 2, the hiker travels 30 km East. Find the total displacement (magnitude and direction) from the point of origin.

21 What are the components of the vector E = A+ D? A. E x < 0, E y < 0 B. E x < 0, E y > 0 C. E x > 0, E y > 0 D. E x > 0, E y < 0 Q1

22 Which is a correct statement about A B? A. x-component > 0, y-component > 0 B. x-component > 0, y-component < 0 C. x-component < 0, y-component > 0 D. x-component < 0, y-component < 0 E. x-component = 0, y-component > 0 Q2

23 Clicker Answers Chapter/Section: Clicker #=Answer Ch.3A:1=A, 2=D

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