Position vectors, Force Vector Directed Along a Line, and Dot Product

Position vectors are important in formulating a Cartesian force vector directed between 2 points in space. The tradition convention of these position vectors is that the z axis points upward.

A position vector r is defined as a fixed vector which locates the point in space relative to another point, which is 

r = xi + yj + zk.

Travels x in the +i direction, y in the +j direction, and z in the +k direction.

Sometimes the position vector is denoted from point A to b and denoted as several subscripts. 

It can be defined as the follows:

r = (XB - XA)i + (YB - YA)j + (ZB - ZA)k

α relates to x, β relates to y, and γ relates to z. 

The next equation that I want to denote here is the force vector directed along a line. It is multiplied by a certain unit vector. 

Here's an example photograph: 

The roof is supported by cables as shown in the photo. If the cables exert forces FAB = 100N and FAC = 120N on the wall hook at A as shown in figure 2-41A, determine the resultant force acting at A. 

To go from A to B, need to travel -4k m then 4i m and to go from A to c, you ust travel -4km, then 4i m, then 2jm.

The following equations ensue.

In statics, one has to find the angle between 2 lines. Dot product is used to multiply the vectors and is a projection of vector A on vector B: 

A o B = AB cos θ

Here are the laws of operations:

Commutative law is A o B = B o A

Multiplication by scalar = a(A o B) = (aA) o B = A o (aB)

The distributive law is A o (B + D) which equals (A o B) + (A o D) 

Also, A o B = AxBx + AyBy + AzBz.

Multiply the corresponding products and sum them up. Here are the applications:

Maybe we want to find the angle between 2 vectors, which θ = cos^-1(A o B / AB)

If the dot product is 0, then A would be perpendicular to B, since the projection is 0.

The components of a vector parallel or perpendicular to a line can also be found, which the component parallel can be defined by Aa = A cos θ, which is the projection of A onto a line, which is 

Aa = A cos θ = A o UA

The scalar projection of A along a line is determined by the dot product of A and a certain unit vector which defines the direction of a line. AA o UA is the component which is represented as a vector. 

A is the parallel add to the perpendicular item, or A = Aa + A⊥. A⊥ = (A^2 - Aa^2)^(0.5).

Now we want to have a pipe question, where a pipe is subjected to force of F = 80lb, and we want to determine the angle θ between F and the pope segment and the projection among this segment. 

So We can determine the following:

and here's the math.