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Vectors


Shodor > CSERD > Resources > Applications > Vectors

  


Vectors

Vectors in science

The word vector comes from the Latin term vehere, to carry. In Biology, a vector is an agent which carries disease, such as a mosquito carrying infected blood from one patient to the next.

In physics, a vector is a quantity which has both a magnitude and a direction associated with it. The most commonly used example of vectors in everyday life is velocity. When you drive your car, your speedometer tells you the speed of your car, but it doesn't tell you where you are going. The combination of both where you are going and how fast you are going there is your car's velocity.

An object's position is also a vector, though this is sometimes confusing. The reason position is a vector is that an object's position must be defined relative to some other position, and in the difference between the object's position and the reference position, we have both a distance and a direction.

In order to express magnitude and direction more formally, we often measure direction as degrees counter clockwise from the x axis, and magnitude as distance from the origin of an xy plane.

This is an image of a plane flying over mountains. The plane radio's `Colorado, what's my position'. The radio tower sends back, `25 miles west southwest'.

Some quantities, on the other hand, do not have any direction associated with them. Temperature, for instance, isn't going anywhere, it just is. Mass is another example of something which does not have a direction associated with it. We call quantities which cannot be expressed as having a direction scalars.

Converting from magnitude and direction to cartesian (x and y) coordinates

To convert from magnitude and direction to x and y coordinates, you simply use a few trigonometric relations.


\begin{displaymath}
x = r cos \theta
\end{displaymath}


\begin{displaymath}
y = r sin \theta
\end{displaymath}

where r is the magnitude, and $\theta$ is the direction measured as an angle counter-clockwise from the positive x-axis.

You should be very careful where you measure your angles from, as this is an easy place to make a mistake.

Some people find it is a bother to work with angles measured fromthe positive x axis when they are working, for example, with vectors that are all pointing in the negative x direction. As long as you set up your problem so that the x and y coordinates are the legs of a right triangle, you can sometimes measure more convenient angles. In this case, you want to use the following form of the cosine and sine equations.


\begin{displaymath}
cos \theta = \frac{adjacent}{hypotenuse}
\end{displaymath}


\begin{displaymath}
sin \theta = \frac{opposite}{hypotenuse}
\end{displaymath}

Either way you get the x and y coordinates, you will likely eventually want to convert back to magnitude and direction. The magnitude is easy, just use the Pythagorean theorem.


\begin{displaymath}
r^2=x^2+y^2
\end{displaymath}

The angle is a little harder. The definition of your sine, cosine, and tangent will help you to get <b>an</b> angle, but not necessarily your angle. The inverse functions may give you an angle which is off by a multiple of $\pi$.

In all cases, you really should draw a diagram before you solve for the direction. This will help you to be sure the answer you get makes sense.


\begin{displaymath}
tan \theta = \frac{y}{x}
\end{displaymath}

For most calculators, if your resultant vector is in quadrant I, this works fine, if it is in quadrant II or III, add $\pi$ to the result, and if it is quadrant IV, add $2 \pi$ to the result.

In 3 dimensions, magnitude and direction are generally known either as x, y, and z coordinates, or magnitude, angle down from the positive z axis ($\phi$) and angle around the x-y plane, counterclockwise from the positive x axis ($\theta$).

In this case


\begin{displaymath}
x = r cos \theta sin \phi
\end{displaymath}


\begin{displaymath}
y = r sin \theta sin \phi
\end{displaymath}


\begin{displaymath}
z = r cos \phi
\end{displaymath}

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