2: Notes

Scalars & Vectors​

Scalar - any positive or negative physical quantity that can be completely specified by its magnitude 
Vector - any physical quantity that requires both a magnitude and a direction for its complete description

The resultant of several coplanar forces can easily be determined if an x, y coordinate system is established and the forces are resolved along the axes
Multiplication or division of a vector by a scalar will change only the magnitude of the vector. 
If vectors are collinear, the resultant is simply the algebraic or scalar addition

Parallelogram Law​

Two forces add according to the parallelogram law. The components form the sides of the parallelogram and the resultant is the diagonal. 

Cartesian Vectors​

Representation​

A vector V is represented by its rectangular components in each of the xyz axes in the form 

V=V_xi+V_yj+V_zk​

Magnitude​

|V|=(V_x²+V_y²+V_z²)^½

Direction​

the direction of V is defined by the coordinate direction angles α, β, and γ measured between the tail of V and the positive x, y, and z axes, respectively

cosα = V_x/V
cosβ = V_y/V
cosγ = V_z/V​

an easy way of obtaining these direction cosines is to form a unit vector U_V in the direction of V

U_V = V/|V| = V_x/V i + V_y/V j + V_z/V k​

an important relation among the direction cosines can be formulated as 

cos²α + cos²β + cos²γ = 1​

Addition​

the addition or subtraction of two or more vectors is greatly simplified if the vectors are expressed in terms of their Cartesian components. A resultant vector R from the addition of two vectors is

V_R = ΣV = ΣV_xi + ΣV_yj + ΣV_zk​

Position Vectors​

a position vector r is defined as a fixed vector which locates a point in space relative to another point
if r extends from the origin of coordinates, O, to point P(x,y,z)

r=xi+yj+zk​

for two vectors r_A and r_B, the position vector represented by the two of them is
r=(x_B-x_A)i+(y_B-y_A)j+(z_B-z_A)k

The easiest way to formulate the components of a position vector is to determine the distance and direction that must be traveled along the x,y,z directions - going from the tail to the head of the vector

A force F acting in the direction of a position vector r can be represented in Cartesian form if the unit vector U of the position vector is determined and it is multiplied by the magnitude of the force, i.e. F=|F|u=|F|(r/|r|)

Dot Product​

The dot product is used to determine the angle between two vectors or the projection of a vector in a specified direction

A⋅B = ABcosθ = A_xB_x + A_yB_y + A_zB_z​

The magnitude of the projection of vector A along a line a whose direction is specified by u_a is determined from the dot product |A_a| = A⋅u_a