VECTORS Vector: a quantity that is fully described by both magnitude (number and units) and direction. Scalar: a quantity that is described fully by magnitude alone. A vector is graphically represented by an arrow whose length reflects the magnitude and whose head reflects the direction. In writing vector equations, vectors are represented by bold faced letters: scalar: a + b = c vector: a + b = c

Notice Vector Direction: In relation to + x axis

Vector Direction: By agreement, vectors are generally described by how many degrees the vector is rotated from the + x axis 30˚ 150˚ Negative 2D vectors: A - A 180˚ opposite

Vectors add trigonometrically, but also follow the commutative and associative laws of algebra: a + b = b + aa + (b + c) = (a + b) + c a - b = a + (-b) b -b A single vector can be resolved into right angle components. These components are traditionally resolved in terms of the x, y and z coordinate plane: x y z

Resolving vectors in 2 dimensions: a  axax ayay a x = a cos  a y = a sin   is always the angle the vector lies off of the +x axis! (from 0˚ to 360˚) The components can (and will) specify the vector: a = √ a x 2 + a y 2 tan  = a y / a x

When resolving a vector, it is conventional to describe the components in terms of unit length with the symbols i, j, and k representing unit vector lengths in the x, y and z directions. in three dimensions: a = a x i + a y j + a z k in two dimensions: 135˚ 17m a = ˚ a x = acos135 = -12 m a y = asin135 = 12 m a = -12 m î + 12 m ĵ

Adding Vectors A vector quantity can be properly expressed (unless otherwise specified) as a magnitude and direction, or as a sum of components. An automobile travels east for 32 km and then heads due south for 47 km. What is the magnitude and direction of its resultant displacement? s = 32km i - 47km j s = √ s x 2 + s y 2 s = √ (-47) 2 = 57 km Ø = tan -1 (s y /s x ) = tan -1 (- 47/32) = -56˚ s = ˚ or 304˚

A woman leaves her house and walks east for 34 m. She then turns 25˚ to the south and walks for 46 m. At that point she head due west for 112 m. What is her total displacement relative to her house? s 1 = s 1x i + s 1y j = 34 m i + 0 j s 2 = s 2x i + s 2y j = 46cos(-25˚)i + 46sin(-25˚)j = 42m i - 19m j s 3 = s 3x i + s 3y j = -112m i + 0j s = s x i + s y j

s x = s 1x + s 2x + s 3x = ( )m = - 36m s y = s 1y + s 2y + s 3y = ( )m = - 19 m s = - 36m i - 19m j If specifically asked for magnitude and direction: s = √ (- 36) 2 + (-19) 2 = 41 m Ø = tan -1 (s y / s x ) = = 208˚ s = ˚

Subtracting Vectors  - Ĉ =  + (-Ĉ) and – Ĉ has the same magnitude as Ĉ but in the exact opposite direction: Ĉ - Ĉ Ĉ = C x î + C y ĵ - Ĉ = -C x î - C y ĵ

Multiplication of Vectors 1) Multiplication of a vector by a scalar: multiply vector a by scalar c and the result is a new vector with magnitude ac, and in the same direction as a. (Ex: F = ma) 2) Multiplication of a vector by a vector to produce a scalar (called the dot product): ab = abcos , where  is the angle between the vectors Work: W = Fr = Fcos  r

3) Multiplication of two vectors to produce a third vector (the cross product): a X b = c where the magnitude of c is defined by c = absin , where  is the angle between the vectors the direction of a X b would be determined by the right hand rule (this will be explained in more depth later) note that a X b would have the same magnitude as b X a, but be in the exact opposite direction! Ex: Torque: T = r x F