Vectors Vectors and Scalars Vector: Quantity which requires both magnitude (size) and direction to be completely specified –2 m, west; 50 mi/h, 220 o.

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Presentation transcript:

Vectors

Vectors and Scalars Vector: Quantity which requires both magnitude (size) and direction to be completely specified –2 m, west; 50 mi/h, 220 o –Displacement; Velocity Scalar: Quantity which is specified completely by magnitude (size) –2 m; 50 mi/h –Distance; Speed

Vector Representation Print notation: A – Sometimes a vector is indicated by printing the letter representing the vector in bold face

Mathematical Reference System x y 0o0o 90 o 180 o 270 o Angle is measured counterclockwise wrt positive x-axis

Equal and Negative Vectors

Vector Addition A + B = C (head to tail method) B + A = C (head to tail method) A + B = C (parallelogram method)

Addition of Collinear Vectors

Adding Three Vectors

Vector Addition Applets Visual Head to Tail Addition Vector Addition Calculator

Subtracting Vectors

Vector Components Horizontal Component A x = A cos  Vertical Component A y = A sin 

Signs of Components

Components ACT For the following, make a sketch and then resolve the vector into x and y components. AyAy AxAx A y = (60 m) sin(120) = 52 m A x = (60 m) cos(120) = -30 m BxBx ByBy B x = (40 m) cos(225) = m B y = (40 m) sin(225) = m

(x,y) to (R,  ) Sketch the x and y components in the proper direction emanating from the origin of the coordinate system. Use the Pythagorean theorem to compute the magnitude. Use the absolute values of the components to compute angle  - - the acute angle the resultant makes with the x-axis Calculate  based on the quadrant*  

*Calculating θ When calculating the angle, 1) Use the absolute values of the components to calculate  2) Compute C using inverse tangent 3) Compute  from  based on the quadrant. Quadrant I:  =  Quadrant II:  = 180 o -  ; Quadrant III:  = 180 o +  Quadrant IV:  = 360 o - 

(x,y) to (R,  ) ACT Express the vector in (R,  ) notation (magnitude and direction) A = (12 cm, -16 cm) A = (20 cm, 307 o )

Vector Addition by Components Resolve the vectors into x and y components. Add the x-components together. Add the y-components together. Use the method shown previously to convert the resultant from (x,y) notation to (R,  ) notation

Practice Problem Given A = (20 m, 40 o ) and B = (30 m, 100 o ), find the vector sum A + B. A = (15.32 m, m) B = (-5.21 m, m) A + B = (10.11 m, m) A + B = (43.6 m, 76.6 o )

Unit Vectors: Notation Vector A can be expressed in several ways Magnitude & Direction (A,  ) Rectangular Components (A x, A Y )