Vectors Vectors and Scalars Vector Quantity which requires

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Vectors

Vectors

Vectors and Scalars • Vector: Quantity which requires both magnitude (size) and direction to

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

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

Mathematical Reference System 90 o y 180 o Angle is measured counterclockwise wrt positive

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

Equal and Negative Vectors

Equal and Negative Vectors

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

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

Addition of Collinear Vectors

Adding Three Vectors

Adding Three Vectors

Vector Addition Applets • Visual Head to Tail Addition • Vector Addition Calculator

Vector Addition Applets • Visual Head to Tail Addition • Vector Addition Calculator

Subtracting Vectors

Subtracting Vectors

Vector Components Vertical Component Ay= A sin Horizontal Component Ax= A cos

Vector Components Vertical Component Ay= A sin Horizontal Component Ax= A cos

Signs of Components

Signs of Components

Components ACT • For the following, make a sketch and then resolve the vector

Components ACT • For the following, make a sketch and then resolve the vector into x and y components. Ay Ax Bx By Ax = (60 m) cos(120) = -30 m Bx = (40 m) cos(225) = -28. 3 m Ay = (60 m) sin(120) = 52 m By = (40 m) sin(225) = -28. 3 m

(x, y) to (R, ) • Sketch the x and y components in the

(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

*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

(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. •

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,

Practice Problem Given A = (20 m, 40 o) and B = (30 m, 100 o), find the vector sum A + B. A = (15. 32 m, 12. 86 m) B = (-5. 21 m, 29. 54 m) A + B = (10. 11 m, 42. 40 m) A + B = (43. 6 m, 76. 6 o)

Unit Vectors: Notation • Vector A can be expressed in several ways • Magnitude

Unit Vectors: Notation • Vector A can be expressed in several ways • Magnitude & Direction (A, ) • Rectangular Components (Ax , AY)