Vectors

Vectors vs Scalars

Vector quantities are those that have a magnitude and a direction. Scalars are quantities that are only expressed with a magnitude. For example, saying that you drove 50 km/h to get to work is a descriptiom of your speed. However, if you say that you drove 50 km/h west to get to work, you have described your velocity. While many people use the words speed and velocity interchageably, they are different. Speed is a scalar - it is only expressed with a magnitude. Velocity is a vector - it is expressed with a magnitude and direction. Temperature, energy, and time are other examples of scalars. Force and momentum are other examples of vector quantities.

Vector Addition - How 3 + 4 =5

If someone were to tell you that 3 + 4 = 5 you would likely be confused as to how that person got that result. However, due to the mathematics of vector addition, this result is not only sensible, but could very well be correct. Consider the diagram below:

This diagram may represent a scenario where a person walks 3 km to the east and then changes direction and walks 4 km to the north. If one wants to calculate this person's displacement (straight line distance from start to finish), a line drawn from the point of origin to the finish can be used to create a right triangle. The displacement vector - the triangle's hypotenuse - can be found using the Pythagorean theorem:
a2 + b2 = c2
- or -
leg2 + leg2 = hypotenuse2
The two legs are referred to as the component vectors of the person's walk. The hypotenuse, the vector that connects start to finish (the displacement) is referred to as the resultant vector.

Solving for a Resultant

Consider the diagram below:

Let's assume that component A has a magnitude of 25 m and component B has a magnitude of 32 m. The resultant magnitude can be found utilizing the Pythagorean theorem:
A2 + B2 = R2
252 + 322 = R2
R2 = 1649
R = 41 m

To determine the value for θ, trigonometric functions must be used. Since the magnitudes of both components and resultant are known, any two sides may be used to find θ, but care must be taken in aligning it with the proper trig function!
sin(θ) = 32/41
sin(θ) = 0.78
sin-1 0.78 = θ
θ = 51°

When magnitude and direction are put together, the resultant is described as 41 m, 51° north of east.