VECTORS. Section 6.3 Precalculus PreAP/Dual, Revised /11/ :41 PM 6.3: Vectors in the Plane 1

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1 VECTORS Section 6.3 Precalculus PreAP/Dual, Revised /11/ :41 PM 6.3: Vectors in the Plane 1

2 DEFINITIONS A. Vector is used to indicate a quantity that has both magnitude (length/distance) and direction. 1. Magnitude Equation: PQ = x 2 x y 2 y Slope: m = y 2 y 1 x 2 x 1 3. Component Form Equation is v = x 2 x 1, y 2 y 1 B. Represented by an arrow or a directed line segment 10/11/ :41 PM 6.3: Vectors in the Plane 2

3 VECTOR EXAMPLE Vector: has magnitude (length) and direction PQ Q x, y 2 2 Terminial Magnitude PQ x x y y Magnitude P x, y Initial /11/ :41 PM 6.3: Vectors in the Plane 3

4 EXAMPLE 1 Find the magnitude of vector PQ where P 2, 3 and Q 5, 9 and its component form. PQ Q 5,9 Terminial P 2,3 Initial 10/11/ :41 PM 6.3: Vectors in the Plane 4

5 EXAMPLE 1 Find the magnitude of vector PQ where P 2, 3 and Q 5, 9 and its component form 2 2 PQ x 5 x 2 y9 y 3 Magnitude /11/ :41 PM 6.3: Vectors in the Plane 5

6 EXAMPLE 1 Find the magnitude of vector PQ where P 2, 3 and Q 5, 9 and its component form Vector/Component Form P Q 2,3 P, P 1 2 5,9 Q, Q 1 2 v Q 1 P v Q P v ,6 10/11/ :41 PM 6.3: Vectors in the Plane 6

7 EXAMPLE 2 Show that u and v are equivalent through magnitude, slope and vector form of v PQ & RS 13 m 2 ; CF : 3,2 3 10/11/ :41 PM 6.3: Vectors in the Plane 7

8 YOUR TURN Find the magnitude of vector PQ where P 3, 5 and Q 7, 11 and component form CF : 10, 16 10/11/ :41 PM 6.3: Vectors in the Plane 8

9 VECTOR OPERATIONS A. Geometrically, the product of a vector v and scalar k is the vector that is k times as long as v B. If k is positive, kv has the same direction as v, and if k is negative, kv has the opposite direction. C. Equivalent Vectors is where every vector is equal to another vector with the initial point at the origin. 10/11/ :41 PM 6.3: Vectors in the Plane 9

10 EXAMPLE 3 If u = 1, 6 and v = 4, 2, solve for u + v and 3v and find the magnitude of u + v u v 1,6 4, 2 3v 3 4,2 v 4,2 O u v 0,0 u 1,6 u v 3,8 3v 12,6 10/11/ :41 PM 6.3: Vectors in the Plane 10

11 EXAMPLE 3 If u = 1, 6 and v = 4, 2, solve for u + v and 3v and find the magnitude of u + v v 4,2 u x y 2 2 u v u 1,6 3 8 u 2 2 u 9 64 O 0, /11/ :41 PM 6.3: Vectors in the Plane 11

12 YOUR TURN If u = 3, 1 and v = 8, 4, solve for 2u 3v and find the magnitude of 2u 3v 2u 3v 30, 10 Magnitude /11/ :41 PM 6.3: Vectors in the Plane 12

13 A. Have a magnitude of 1 B. Solve for the magnitude FINDING THE UNIT VECTOR C. Unit Vector equation: u = v = v v x 2 +y 2 D. To find a unit vector u that has the same direction as vector v: u = v = x 1 v v, y 1 v 10/11/ :41 PM 6.3: Vectors in the Plane 13

14 Find the unit vector v = 2, 5 EXAMPLE 4 v v v x y 2 2 v v 2, , , u , /11/ :41 PM 6.3: Vectors in the Plane 14

15 EXAMPLE 4 Find the unit vector v = 2, v u v v , /11/ :41 PM 6.3: Vectors in the Plane 15

16 Find the unit vector of v = 7, 3 EXAMPLE 5 u , /11/ :41 PM 6.3: Vectors in the Plane 16

17 Find the unit vector of v = 6, 1 YOUR TURN u , /11/ :41 PM 6.3: Vectors in the Plane 17

18 LINEAR COMBINATION/ALTERNATE NOTATION A. Vectors v = v 1, v 2 is also represented as v = i + j B. i and j are considered component vectors whereas xi + yj is a linear combination of i and j 10/11/ :41 PM 6.3: Vectors in the Plane 18

19 EXAMPLE 6 The initial point of a vector is 0, 2 and the terminal point is 3, 6. Write a linear combination of the standard vector. u x x, y y ,6 2 3,8 u 3i 8j 10/11/ :41 PM 6.3: Vectors in the Plane 19

20 YOUR TURN The initial point of a vector is 2, 6 and the terminal point is 8, 3. Write a linear combination of the standard vector. u 6i 3j 10/11/ :41 PM 6.3: Vectors in the Plane 20

21 DIRECTIONAL ANGLES A. Direction Angle: measured counterclockwise from the x axis to terminal point of u B. u = x, y = cos θ, sin θ = cos θ i + sin θ j C. v = v = cos θ, sin θ = v cos θ i + v sin θ j D. Since v = ai + bj, tan θ = b and use reference angles and Unit a Circle when possible (with the exception of Quadrant III) E. When finding angles, remember the reference angle rules 10/11/ :41 PM 6.3: Vectors in the Plane 21

22 EXAMPLE 7 Find the direction angle and magnitude of u = 3i + 3j Tan a 1 b Tan /11/ :41 PM 6.3: Vectors in the Plane 22

23 EXAMPLE 8 Find the direction angle of u = 3i 4j 1 4 Tan 3 Tan 4 4 Tan /11/ :41 PM 6.3: Vectors in the Plane 23

24 YOUR TURN Find the direction angle of v = 4i 5j /11/ :41 PM 6.3: Vectors in the Plane 24

25 EXAMPLE 9 Find the component form of a vector that represents the velocity of an airplane descending at a speed of 150 miles per hour at an angle of 20 below the horizontal. v v cosi v sin j v 150 cos 200 i 150 sin 200 j v i j , /11/ :41 PM 6.3: Vectors in the Plane 25

26 YOUR TURN Find the component form of the velocity vector that represents an airplane descending West at a speed of 100 mph at an angle of 30 below horizontal , 50 10/11/ :41 PM 6.3: Vectors in the Plane 26

27 MORE RESULTANT FORCE STEPS A. The wind pushes it in another direction. That is another vector. B. The resultant vector is the ground speed. C. Identify the two forces and then add them D. u = ai + bj represents the vector where v cos θ i + v sin θ j E. Identify the magnitude and direction using the appropriate equations 10/11/ :41 PM 6.3: Vectors in the Plane 27

28 EXAMPLE 10 A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 mph tail wind acting in the direction of 60 north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N v 60 o u E 10/11/ :41 PM 6.3: Vectors in the Plane 28

29 EXAMPLE 10 A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 mph tail wind acting in the direction of 60 north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? Need to find the magnitude and direction of the resultant vector u + v. 10/11/ :41 PM 6.3: Vectors in the Plane 29 N v u + v u E

30 EXAMPLE 10 A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 mph tail wind acting in the direction of 60 north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? u v 500,0 70cos60,70sin 60 v 35,35 3 uv 535,35 3 u v u v ; /11/ :41 PM 6.3: Vectors in the Plane 30 tan

31 EXAMPLE 10 A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70 mph tail wind acting in the direction of 60 north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N v o 6.5 o u E 10/11/ :41 PM 6.3: Vectors in the Plane 31

32 ASSIGNMENT Page odd, odd (DO NOT SKETCH), odd, odd, odd 10/11/ :41 PM 6.3: Vectors in the Plane 32