Mechanics: Scalars and Vectors

 Scalars can be 100% specified with magnitude. Vectors require both direction and magnitude. The following represents vector multiplication:

A multiplication switches direction if the factor is negative and remains the same if the direction is positive. Then the vector is scaled by a certain factor. 

The parallelogram and triangle vector addition method is enumerated here:

When Vectors A and B are added, these 2 vectors are pointed from the same origin, and a "parallelogram" is made using this addition. We can also have the head of one vector point to the tail of the other, and the addition is from the tail pointing to the head. 

Now the vector subtraction is extremely simple, just add the negative of a vector, as follows: 

The following are examples asking to determine the magnitude of certain resulting forces. 

The next question involves an eyebolt. We need a resultant force acting on the eyebolt, and that F2 have a minimum magnitude. What is the minimum magnitude of this for straightness? We want to find a magnitude when F2 line of action is perpendicular to the line of action FR. The following equations ensue.

The next topic that I want to go over is the addition of a system of coplanar forces, when a force is resolved into 2 components onto the x and y aces, the components are called regtangular components. We can put force F into a rectangle such that the following: 

The y component can also be a negative scalar, especially if it points down. The head of the vector arrow in any figure indicates the sense of the vector graphically. 

We can express the vector F and Cartesian vector F = Fxi + Fyj.

Here's a diagram using Cartesian vector notation, and representing each vector together:

And then we can determine the magnitude an tangent of FR as follows:

The orientation of the x and y axes is arbitrary.

We can greatly simplify vector operations, as long as we first represent them in Cartesian Vector form, and there is a general method for doing this, and a method for finding the resulting force of a system of concurrent forces. The first thing that we want to do is the right hand rule, if the right hand thumb points towards the x axis, from the x axis to the y axis. 

A vector A can have up to 3 rectangular components. We can represent components as A = A' + Az then A' = Ax + Ay. combining these, we can get that A = Ax + Ay + Az. 

X is i, Y is j, and Z is k. 

Now we can represent A as Axi + Ayj + Azk, and it's good to separate the magnitude and direction of each of these component vectors. This we can get

A = (Ax^2 + Ay^2 + Az^2)^(1/2)

Now we want to determine the direction of a Cartesian vector, denoted by alpha, beta and gamma. 

We get that cos(α) = Ax/A, cos(β) = Ay/A, and cos(γ) = Az/A.

Now we can represent the unit vector uA as the following:

UA = Ax/A i + Ay/A j + Az/A k. Where cos2α + cos2β + cos2γ = 1. And if A is the magnitude, then A = AUA.

The direction of the vector A can also be specified using the angles θ and φ as Az = A cos φ and A' = A sin φ, which means Ax = A sin φ cos θ  and Ay = A sin φ sin θ. Written in final Cartesian vector form, A = A sin φ cos θ i  + A sin φ sin θ j  + A cos  φ k.

Another aspect here is the addition of Cartesian Vectors. The following equation pertains if we want a vector sum of all the forces: 

FR = ΣF = ΣFxi + ΣFyj + ΣFzk.

Here's a sample problem involving these vectors: 

The problem is how we express a force F in a certain Cartesian Vector, and a solution is denoted below:

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