Force is a vector and is a mechanical disturbance or a load, it candeform the object, change its motion, or both (think of kicking a football)

(line of action, direction)

Length of arrow isproportional to themagnitude if thereis more than one force

Summary of vector operations

Addition

negative of any vector is avector of same magnitudepointing in the opposite direction

C Resultant

3 coplanar vectors

Tip to tail method

Parallelogram method, coplanar

s is a line onto which A isprojected, As is the projectionof A onto line s, noteThat As = A cos

Resolution of Vectors onto the x and y axis of a 2 D system, the reverseof adding two vectors by parallelogram method

The components of the vectorA are Ax and Ay and they actIn the x and y directions, theyAre given by:

Ax = A cos

Ay = A sin

Unit vectors have a magnitude of 1

Note that the book’s vectorterminology is such that anunderlined variable denotes avector and if the name is notunderlined that is the magnitude of the vector

Unit coordinate vectors

The letters i , j , k are commonlyused to denote the unit coordinate vectorsin the x, y, z directions of a Cartesian coordinate system

So a vector like A can be writtenIn terms of these unit vectors and theComponents Ax, Ay, Az as

A = Ax i + Ay j + Az z

Trig method for adding vectorsSteps:1. First resolve each vector into its component parts where

for example Ax = A cos and Ay = A sin

2. Then we can write thatA = Ax i + Ay j

= A cos i + A sin j

3. Magnitude of A is also givenby A = ( Ax2 + Ay2 )1/2

4. Likewise for vector B we have

B = B cos i + B sin j

5. C = A + B so we have that

C = (Acos + Bcos) i + (Asin + Bsin) jHence Cx = (Acos + Bcos) Cy = (Asin + Bsin)

6. C = ( Cx2 + Cy2 )1/2

= tan-1 (Cy / Cx)

Dot or scalar product is defined as theproduct of the magnitude of two vectorsmultiplied by the cosine of the angle between them

A B = A B cos

Conceptually it is the same as projectingvector B onto the line of action of vectorA, ie. Ba = B cos then multiplying thisby the magnitude of A

What is:1. i i = ?

2. i j = ?

So to take the dot or scalar product of two vectors A and B we proceed asfollows below:

After recognizing these very important unit coordinate vector identities:

Vector or cross

A x B = C, where the vector Chas a magnitude equal to theproduct of the magnitudes of Aand B times the sine of the angle ,

So we can write C = A B sin

Note that vector C has a direction thatIs perpendicular to the plane definedBy vectors A and B, so in this exampleC is coming out at us along the z axis

vector

Sense is based on the right hand rule

Note a vector !