Statics/Force Vectors

Many quanities in mechanics are vectors. A vector is a quantity that has a magnitude and a direction. Velocity is a vector. To solve a problem, it is not enough to know the speed a vehicle is travelling, but one must also know the direction in which the vehicle is travelling at that speed. Likewise, Force is a vector quantity. It has a magnitude (the amount of force in Newtons) and the direction in which the force is being applied.

Graphical Representation

Vectors are usually denoted in boldface, as a. Other conventions include ${\vec {a}}$ or a, especially in handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type.

Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below:

Vector arrow pointing from A to B

Here the point A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction.

In the figure above, the arrow can also be written as ${\overrightarrow {AB}}$ or AB.

On a two-dimensional diagram, sometimes a vector perpendicular to the plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre indicates a vector pointing out of the front of the diagram, towards the viewer. A circle with a cross inscribed in it indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip an arrow front on and viewing the vanes of an arrow from the back.

A vector in the Cartesian plane, with endpoint (2,3). The vector itself is identified with its endpoint.

Dot Product

The dot product of two vectors a = [a₁, a₂, … , a_n] and b = [b₁, b₂, … , b_n] is defined as:

$\mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$

where Σ denotes summation notation and n is the dimension of the vectors.

Scalars

Magnitudes without directions are referred to as scalars. These are not vectors but only single numbers. scalar have only magnitude