7.1 Vectors and Direction 1

Chapter 7 Objectives Add and subtract displacement vectors to describe changes in position. Calculate the x and y components of a displacement, velocity, and force vector. Write a velocity vector in polar and x-y coordinates. Calculate the range of a projectile given the initial velocity vector. Use force vectors to solve two-dimensional equilibrium problems with up to three forces. Calculate the acceleration on an inclined plane when given the angle of incline. 2

Chapter 7 Vocabulary Cartesian coordinates component cosine displacement inclined plane magnitude parabola polar coordinates projectile Pythagorean theorem range resolution resultant right triangle scalar scale sine tangent trajectory velocity vector x-component y-component

Inv 7.1 Vectors and Direction Investigation Key Question: How do you give directions in physics? 4

7.1 Vectors and Direction A scalar is a quantity that can be completely described by one value: the magnitude. You can think of magnitude as size or amount, including units. 5

7.1 Vectors and Direction A vector is a quantity that includes both magnitude and direction. Vectors require more than one number. The information “1 kilometer, 40 degrees east of north” is an example of a vector. 6

7.1 Vectors and Direction In drawing a vector as an arrow you must choose a scale. If you walk five meters east, your displacement can be represented by a 5 cm arrow pointing to the east. 7

7.1 Vectors and Direction Suppose you walk 5 meters east, turn, go 8 meters north, then turn and go 3 meters west. Your position is now 8 meters north and 2 meters east of where you started. The diagonal vector that connects the starting position with the final position is called the resultant. 8

7.1 Vectors and Direction The resultant is the sum of two or more vectors added together. You could have walked a shorter distance by going 2 m east and 8 m north, and still ended up in the same place. The resultant shows the most direct line between the starting position and the final position. 9

7.1 Representing vectors with components Every displacement vector in two dimensions can be represented by its two perpendicular component vectors. The process of describing a vector in terms of two perpendicular directions is called resolution.

7.1 Representing vectors with components Cartesian coordinates are also known as x-y coordinates. The vector in the east-west direction is called the x-component. The vector in the north-south direction is called the y-component. The degrees on a compass are an example of a polar coordinates system. Vectors in polar coordinates are usually converted first to Cartesian coordinates.

7.1 Adding Vectors Writing vectors in components make it easy to add them.

7.1 Subtracting Vectors To subtract one vector from another vector, you subtract the components.

Calculating the resultant vector by adding components An ant walks 2 meters West, 3 meters North, and 6 meters East. What is the displacement of the ant? You are asked for the resultant vector. You are given 3 displacement vectors. Sketch, then add the displacement vectors by components. Add the x and y coordinates for each vector: X1 = (-2, 0) m + X2 = (0, 3) m + X3 = (6, 0) m = (-2 + 0 + 6, 0 + 3 + 0) m = (4, 3) m The final displacement is 4 meters east and 3 meters north from where the ant started.

7.1 Calculating Vector Components Finding components graphically makes use of a protractor. Draw a displacement vector as an arrow of appropriate length at the specified angle. Mark the angle and use a ruler to draw the arrow.

7.1 Finding components mathematically Finding components using trigonometry is quicker and more accurate than the graphical method. The triangle is a right triangle since the sides are parallel to the x- and y-axes. The ratios of the sides of a right triangle are determined by the angle and are called sine and cosine.

7.1 Finding the Magnitude of a Vector When you know the x- and y- components of a vector, and the vectors form a right triangle, you can find the magnitude using the Pythagorean theorem.

Finding two vectors… Robots are programmed to move with vectors. A robot must be told exactly how far to go and in which direction for every step of a trip. A trip of many steps is communicated to the robot as series of vectors. A mail-delivery robot needs to get from where it is to the mail bin on the map. Find a sequence of two displacement vectors that will allow the robot to avoid hitting the desk in the middle? You are asked to find two displacement vectors. You are given the starting (1, 1) and final positions (5,5) Add components (5, 5) m – (1, 1) m = (4, 4) m. Use right triangle to find vector coordinates x1 = (0, 4) m, x2 = (4, 0) m Check the resultant: (4, 0) m + (0, 4) m = (4, 4) m