Chapter 4 Vectors (4.1) Determine graphically the sum of two or more vectors. Establish a coordinate system in problems involving vector quantities. Use the process of resolution of vectors to find out the components of vectors Determine algebraically the sum of two or more vectors by adding the components of the vectors.

Representing Vector Quantities Vectors have both magnitude and direction, magnitude is always positive, direction can be + or - Two types: Graphical…drawing arrows Always draw tail to tip, utilize n,s,e,w coordinates. Algebraic…d = 50 km southwest, or 10 m east The resultant A resultant vector is a vector that is equal to the sum of two or more vectors. Always draw resultant from tail of first arrow to tip of last. + = A B Or Resultant + = A B

Finding the magnitude of a resultant Graphically, the magnitude of the resultant can be found with the Pythagorean theorem R2 = A2 + B2 ? 6 10

Relative velocities: (draw these situations in notes) Many situations involve two velocities, for example, if you are flying in a plane traveling east at 400 m/s and while on the plane you walk to the west at 2 m/s, what is your speed relative to the plane? Relative to the ground? Likewise if you are aboard a bus traveling 15 m/s to the north and you are walking north at 2 m/s while on the bus what is your speed relative to the bus? Relative to the ground?

Relative velocities: Many times these problems involve boats on a river or plane with a cross wind. Whenever you analyze such a problem draw two vectors, one for the velocity of water or wind, the other for the velocity of the boat/plane. V air relative to ground V plane relative to air V plane relative to ground

The components lie on x and y axis where: Vector Components (4.2) Vector’s can be broken down into components. For example vector A can be broken into a component that lies in the x direction and one that lies in the y direction giving you components: Ax and Ay The components lie on x and y axis where: + x is east + y is north x is west - y is south Ay Ɵ Direction (Ɵ) is typically assigned in degrees, going counterclockwise from east. Ax

Finding magnitude of a vector from components Depends on what side you are given and what you are trying to find. 40 side Ax A side Ay 40°

Using Trig to find sides If given the two components Ax and Ay use Pythagorean theorem If given the hypotenuse (resultant), the direction in degrees and trying to find one of the components use one of the trig functions Sin or Cos…the formulas are Ax = hyp cos Ɵ Ay = hyp sin Ɵ

Practice A 10 kg box is being pulled by a rope with a 15 N force at 30° relative to the horizontal. What is the magnitude of force pulling the box to the east? 15 N 8 N 30°

Algebraic addition of vectors Two or more vectors may be added by first resolving each into its x and y components. The x components are added Rx = ax + Bx + Cx + … The y components are added Ry = Ay + By + Cy + …

Finding direction of a resultant The direction (Ɵ) of a resultant can be found using the formula Ɵ = sin-1 (Ay/hyp) Ɵ = cos-1 (Ax/hyp) Ɵ = tan-1 (Ay/Ax)

Practice An airplane is flying at 250 km/hr towards the south. It encounters a crosswind from the west with a magnitude of 15 km/hr. What is the planes resulting velocity?