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˚ 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 + a a + (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:

xy

z

Resolving vectors in 2 dimensions:

a

ax

ay

ax = a cos

ay = 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 = √ ax2 + ay

2 tan = ay / ax

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 = axi + ayj + azk

• in two dimensions:

135˚17m

a = 17 m @ 135˚

ax = acos135 = -12 m

ay = asin135 = 12 m

a = -12 m î + 12 m ĵ

Adding VectorsA 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 = √ sx2 + sy

2

s = √ 322 + (-47)2

= 57 km

Ø = tan-1 (sy/sx)

= tan-1 (- 47/32) = -56˚

s = 57 km @ -56˚ 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?

s1 = s1xi + s1yj = 34 m i + 0 j

s2 = s2xi + s2yj = 46cos(-25˚)i + 46sin(-25˚)j= 42m i - 19m j

s3 = s3xi + s3yj = -112m i + 0j

s = sxi + syj

sx = s1x + s2x + s3x = (34 + 42 - 112)m = - 36m

sy = s1y + s2y + s3y = (0 - 19 + 0)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 (sy / sx)

= - 19

- 36

= 208˚

s = 41 m @ 208˚

Subtracting Vectors

 - Ĉ =  + (-Ĉ) and – Ĉ has the same magnitude as Ĉ but in the exact opposite direction:

Ĉ

- ĈĈ = Cxî + Cy ĵ

- Ĉ = -Cxî - Cy 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):

a•b = 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