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Let's suppose that we are on a street corner and we see several cars coming in different directions. All of the speedometers read 10[m/s] then should we say they have the same velocity? Usually we would say yes, but let us consider that two cars are put facing towards each other separated by a distance of 40 meters and they both run towards each other, both at a speed of 10 m/s. We would usually think that they would collide in 4 seconds, but they collide in 2 seconds, because they are running towards each other, so even though they have the same speedometer reading they don't have the same velocity. This is why we consider that two particles in motion have the same velocity only if they both run in the same direction at the same speed. From that analysis we can say that velocity is a vector. A vector is a magnitude or measure which has a magnitude (speed) or number figure and a direction. Examples of vectorial magnitudes are: speed, acceleration, force, etc. and examples of non-vectorial magnitudes or scalar magnitudes are: temperature, time and any other that indicates only quantity. Representing Vectors There are two ways to represent vectors: graphically in a Cartesian system of coordinates and literally by expressing the measure of each of its components (this will be explained later because it will be easier once the representation in a system of coordinates is understood). The following is a representation of a vector in a system of coordinates:
This is the representation of a vector named "A" (we will always use capital letters to name vectors), another way to name vectors is to name the point of origin (i.e. O) and the point where the vector ends (i.e.A), then we would call the vector OA. The Greek letter a is the angle between the horizontal and the vector, Greek letters are often used to represent angles. there are several manners to find a vector's magnitude, mostly using triogonometrical functions because the vector and the axes always form two triangles that have an angle of 90°. We will use the same formula we used to take measures in the Cartesian systems of coordinates. We will call the module A, the same name as the vector. The first step is to determine the vector's components on each axis. In the example above, we can see that the vector's component in x is 2 units, and its component in y is 3. Knowing that we can proceed to the formula:
Once knowing the module, we can calculate a with trigonometrical functions, and if we wouldn't know any of the measures of the components we could also calculate them with trigonometrical functions. If you want to know more about this click here. To represent a vector literally we just mention its components in a parenthesis, so in our example the literal representation would be: (2,3). The first number represents the component on the x axis, and the second one the component on the y axis.
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