Presentation on theme: "Starter If the height is 10m and the angle is 30 degrees,"— Presentation transcript:
1 Starter If the height is 10m and the angle is 30 degrees, how long is the shadow?h/s = tanq, so s =h/tanq = 10/tan30 = 17.3m
2 Vectors and Vector Addition 1. Characteristics of Vectors2. Multiplying a vector by a scalar3. Adding Vectors Graphically4. Adding Vectors using Components
3 What is a vector?A vector is a mathematical quantity with two characteristics:1. Magnitude or Length2. Direction ( usually an angle)
4 Vectors vs. ScalarsA vector has a magnitude and direction. Examples: velocity, acceleration, force, torque, etc.
5 Vectors vs. ScalarsA scalar is just a number. Examples: mass, volume, time, temperature, etc.
6 A vector is represented as a ray, or an arrow. The terminal endor headVThe initial end or tail
7 Picture of a Vector Named A Magnitude of AA = 10Direction of Aq = 30 degrees
8 The Polar Angle for a Vector Start at the positive x-axisand rotate counter-clockwiseuntil you reach the vector.That’s how you find thepolar angle.
9 Two vectors A and B are equal if they have the same magnitude and direction. This property allows us to move vectors aroundon our paper/blackboard without changingtheir properties.
10 A = -B says that vectors A and B are anti-parallel A = -B says that vectors A and B are anti-parallel. They have same size but the opposite direction.A = -B also impliesB = -ABA
11 Graphical Addition of Vectors ( Head –to Tail Addition ) To find C = A + B :1st Put the tail of B onthe head of A.2nd Draw the sum vectorwith its tail on thetail of A, and its headon the head of B.Example: If C = A+B, draw C.Here’s Vector C
12 Graphical Addition of Vectors ( Head –to Tail Addition ) To find C = A - B :1st Put the tail of -B onthe head of A.2nd Draw the sum vectorwith its tail on thetail of A, and its headon the head of -B.Example: If C = A-B, draw C.Here’s Vector C = A - B
13 Addition of Many Vectors BBDCCRDAdd A,B,C, and DR = A + B + C + D
16 A vector A in the x-y plane can be represented by its perpendicular components called Ax and Ay. Components AX and AYcan be positive, negative,or zero. The quadrantthat vector A lies indictates the sign of thecomponents.Components are scalars.AAYxAX
17 When the magnitude of vector A is given and its direction specified then its components can be computed easilyyAX = AcosqAAYAY = AsinqxAXYou must use the polar angle in these formulas.
18 Example: Find the x and y components of the vector shown if A = 10 and q = 225 degrees. AX = Acosq = 10 cos(225)=Ay = Asinq = 10 sin(225)A = (-7.07, -7.07)
19 The magnitude and polar angle vector can be found by knowing its components = tan-1(AY/AX) + C
20 Example: Find A, and q if A = ( -7.07, -7.07) = 10 = tan-1(AY/AX) + C= tan-1(-7.07/-7.07) + 180= 225 degrees
21 Example: Find A, and q if A = ( 5.00, -4.00) = 6.40 = tan-1(AY/AX) + C= tan-1(-4.00/5.00) + 360= 321 degrees
22 A vector can be represented by its magnitude and angle, or its x and y components. You can go back and forth’from each representation with these formulas:If you know Ax and Ayyou can get A and q with:If you know A and q, you can get Ax and Ay with:Ax = AcosqAy = Asinq
23 Adding Vectors by Components If R = A + BThen Rx = Ax + Bxand Ry = Ay + BySo to add vectors, find their components and add the like components.
24 ExampleA = ( 3.00,2.00) and B = ( 0, 4.00)If R = A + B find the magnitude and direction of R.Solution: R = A + B = ( 3.00,2.00) + ( 0, 4.00),so R = ( 3.00, 6.00)Then R = ( )1/2 = 6.70q = tan-1( 6/3) = 63.4o
25 Example If R = A + B find the magnitude and direction of R. 1st: Find the components of A and B.Ax = 10cos 30 = 8.66Ay = 10 sin30 = 5.00Bx = 8cos 135 = -5.66By = 8sin 135 = 5.662nd: Get Rx and RyRx = Ax + Bx = = 3.00Ry = Ay + By = = 10.73rd: Get R and q : R = ( )1/2 = 11.1q = tan-1 ( 10.7/3.00) = 74.3o
26 Summary If R = A + B Rx = Ax + Bx Ax = Acosq Ay = Asinq Ry = Ay + By If you know Ax and Ayyou can get A and q with:If you know A and q, you can get Ax and Ay with:Ax = AcosqAy = AsinqIf R = A + B Rx = Ax + BxRy = Ay + By