Vectors in Two Dimensions Read Chapter 1.6-1.9

Scalars and Vectors Scalar quantities have only magnitude. All measurements are considered to be quantities. In physics, there are 2 types of quantities – SCALARS AND VECTORS. Scalar quantities have only magnitude. Vector quantities have magnitude and direction. time mass temperature displacement Magnetic Field acceleration Force velocity Gravitational Field

Vectors are used to describe motion and solve problems concerning motion. For this reason, it is critical that you have an understanding of how to represent vectors add vectors subtract vectors manipulate vector quantities. in 2 and 3 dimensions.

Vectors tip 8 units 2 units 5 units tail Magnitude represented by the length of the vector

Vectors y 1200 800 00 2250 300 600 from -x 3000 x -600 450 from-y Direction represented by the direction of the arrow

Adding Vectors - + Dx1 + Dx2 = +4 mi + (-7 mi) = -3 mi We know how to add vectors in 1-dimension. Example: displacement. If someone walks 4 mi east (Dx1) and then 7 mi west (Dx2) the total displacement (Dx1 + Dx2) is 3 mi west. 1. Adding vectors GRAPHICALLY – place them TAIL TO TIP Dx1 + Dx2 = 3 mi Dx1 = 4 mi west - east + Dx2 = 7 mi 2. Adding vectors MATHEMATICALLY – In 1- dimension, assign direction + or – and add algebraically Dx1 + Dx2 = +4 mi + (-7 mi) = -3 mi

Adding Vectors What about if the vectors are in different directions in 2-D? How do we describe the direction? How do we add/subtract the vectors? v1 = 5 35o below +x v2 = 3 50o above +x

Adding Vectors Graphically “tail to tip” To add vectors using the tail to tip method Draw the first vector (7u, 50o N of E) beginning at the origin. Draw the second vector (3u, 35o S of E) with its tail at the tip of the first vector. Draw the Resultant vector (the answer) from the tail of the first vector to the tip of the last. North South East West v2 v1 v = v1 + v2

Adding Vectors Grahically “parallelogram method” To add vectors with the parallelogram method Draw the first vector to scale beginning at the origin. Draw the second vector, to scale, with its tail also at the origin. Starting at the tip of one vector, draw a dotted line parallel to the other vector. Repeat, starting from the tip of the second vector. Draw the Resultant vector (the answer) from the origin to the intersection of the dotted lines. North South East West v1 v = v1 + v2 v2

Adding Vectors Mathematically North South East West What if the vectors are in different directions? For example, what if I walk 5 steps north and then 4 steps east. What is my total displacement , Dx, for the trip? OR what is the vector sum of the Dx1 and Dx2? Dx2 =4 steps east Dx1= 5 steps north Dx = Dx1+Dx2 = ?

Adding Vectors Mathematically 4 steps east 5 steps north Dx = ? q Use Pythagorean Theorem to find the magnitude of Dx Use right triangle trig to find the direction of Dx

RIGHT TRIANGLE TRIGONOMETRY A – side adjacent to angle q O – side opposite to angle q H – hypotenuse of triangle Pythagoreon Theorem SOHCAHTOA H sinq = O H O cosq = A H q tanq = sinq cosq = O A A

R = 283 m R2 = A2 +B2 = 2402 + 1502 tanq = opp adj 150m 240m = A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip? North MAGNITUDE R2 = A2 +B2 = 2402 + 1502 R = 283 m DIRECTION A = 240 m West East tanq = opp adj 150m 240m = q B= 150m A R=283 m tanq = 0.625 = tan-1(1.3) = 32o south of east Notice that A and B are perpendicular components of R. They are the amount of R in each direction South

The net displacement is 283 m in the direction of 32o S of E. A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip? North The net displacement is 283 m in the direction of 32o S of E. The average velocity: 240 m West East q v = Dx Dt 283 m 0.5 hr 150m = 283 m = 566 m/hr in the direction 32o S of E. South

R = 9.2 u R2 = A2 +B2 = 82 + 4.52 tanq = opp adj 4.5 u 8 u = EXAMPLE: What is the Resultant of adding 2 vectors, A and B, if A = 8 units south and B = 4.5 u west? MAGNITUDE R2 = A2 +B2 = 82 + 4.52 North R = 9.2 u DIRECTION tanq = opp adj 4.5 u 8 u West East = tanq = 0.56 q-1 = tan-1(0.56) = 29o 9.2 u 8u south R q The Resultant is 9.2 units in the direction of 29o south of west South 4.5u west Notice that A and B are perpendicular components of R. They are the amount of R in each direction

Perpendicular Components of a Vector Any vector can be resolved into perpendicular components. Use right triangle trig – x and y components always make a right triangle with the vector . Ex. Vector A has magnitude 8.0 m at an angle of 30 degrees below the x-axis. What are the x- and y-components of A? cos30 = Ax A Ax = Acos30 Ax Ax = 8cos30 = 8(0.866) = +6.9 30o Ay A=8 sin30 = Ay A Ay = Asin30 Ay = 8sin30 = 8(0.5) = - 4m You must assign the correct direction

x- and y- Components of a Vector What are the x- and y- components of the vector A, shown below? y A = 8 m Ay Ax = Acosq = 8cos(30) = 8(0.866) = -6.9 m 30o x Ax Ay = Asinq = 8sin(30) = 8(0.5) = +4 m Ax is the amount of A in the x-direction Ay is the amount of A in the y-direction

A butterfly moves with a speed of 12 m/s A butterfly moves with a speed of 12 m/s. The x-component of its velocity is 8.00 m/s. The angle between the direction of its motion and the x-axis must be what? North v = 12 m/s vy q ? West cosq = vx v 8 12 = vx = 8 East q = cos-10.667 = 48o South

Adding Vectors by Components What about adding 2 vectors, A and B, that are NOT perpendicular or parallel? A + B = R Bx y (Ax+Ay)+(Bx+By)=(Rx+Ry) By B Any vector can be described as the sum of perpendicular components. 60o Components in the same direction are added as 1-D vectors to find the components of the resultant vector. Ry R A Rx = Ax + Bx = 0 + Bx x Rx Ry = Ay + By

Adding x- and y- Components of a Vector A + B = R (Ax+Ay)+(Bx+By)=(Rx+Ry) y Bx Determine the perpendicular components of each vector. Make a table to add up x and y components separately: By B 60o x Y A Ax Ay B Bx By R Rx Ry R Ry A q Rx x Magnitude of R: R2 = Rx2 + Ry2 Direction of R:

ADDING VECTORS GRAPHICALLY You can add as many vectors as you want North A + B + C + D = R R D Graphically, the vectors are added “tail to tip” East West A and the order doesn’t matter C B C + A + D+ B = R South

A + B + C + D = R D A C B Mathematically ADDING VECTORS (Ax+Ay)+(Bx+By)+(Cx+Cy)+(Dx+Dy)=(Rx+Ry) To Add Vectors by Components: Place each vector at origin. Find the x- and y- components of each vector in the sum, and list them in a table. Make sure to include the direction of each component by hand. Add all the x-components to find Rx. Add all the y-components to find Ry. Use Pythagorean Theorem and trigonometry to find the magnitude and direction of R. x Y A -Ax -Ay B +Bx -By C +Cx +Cy D -Dx +Dy R Rx Ry D A R2 = Rx2 + Ry2 Magnitude of R: C Direction of R: B

EXAMPLE (Adding Vectors by Components) Determine the resultant of the following 3 displacements: A. 24m, 30º north of east B. 28m, 37º east of north C. 20m, 50º west of south Bx North B A x (m) y (m) A 20.8 12.0 B 16.9 22.4 C -15.3 -12.9 Ʃ 21.5 By 37o Ay West 30o Ax East 50o Cy C Cx South South

R = 31 m, 440 N of E (from the x-axis) EXAMPLE R North x (m) y (m) A 20.8 12.0 B 16.9 22.4 C -15.3 -12.9 Ʃ R 21.5 Ry q West Rx East South Magnitude Direction R = 31 m, 440 N of E (from the x-axis) South

DO NOW An airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement? x (km) y (km) A 620 B 311 -311 C -331 -439 DR 600 -750 North A Cx West Bx East 45o By 53o Cy B C South DR

DR = 960 km, -510 from the x-axis (510 S of E) EXAMPLE (cont.) North x (km) y (km) A 620 B 311 -311 C -331 -439 DR 600 -750 West East q DR South Magnitude Direction DR = 960 km, -510 from the x-axis (510 S of E)

DR EXAMPLE q 960 km at 510 South of East An airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement? North West East q DR South 960 km at 510 South of East

- + = = - -v2 -v2 Subtracting Vectors v1 v2 v1 v1 v2 v1 In order to subtract a vector, we add the negative of that vector. The negative of a vector is defined as a vector in the OPPOSITE direction (with each component the negative of the original) v1 v2 - v1 -v2 + = -v2 = v1 v2 - v1

Graphical Representation B=10 60o -B 60o A + B = S A – B = D -B A D Tail to tip Tail to tip (A and –B) A B A B S Tail to tail (A and B) D

- 5 + 8.7 Mathematical Representation A=8 B=10 A + B = S A + (–B) = D x y A + 8 B + 5 - 8.7 S 13 -8.7 x y A + 8 -B - 5 + 8.7 D 3 +8.7 -B 60o S Sx Sy q D Dx Dy q

Scalar Multiplication Multiplication of a vector by a positive scalar changes the magnitude of the vector, but leaves its direction unchanged. The scalar changes the size of the vector. The scalar "scales" the vector. Multiplication of a vector by a negative scalar changes the magnitude of the vector, and makes the direction opposite. Example: 3A = 3 x = = If the scalar is negative, it changes the direction of the vector. -3A = -3 x = =