One useful property of vectors is that they can be broken down into components along the axes of a coordinate system using unit vectors.
Consider a position vector \(\vec{r}\) shown in the two-dimensional Cartesian coordinate system below.
Figure1.6.1.A representation of a position vector in an \(xy\)-coordinate system showing the positive angle from the \(x\)-axis and the components of \(\vec{r}\) along the \(x\)- and \(y\)-axes .
You can write \(\vec{r}\) in terms of its components in the \(x\)- and \(y\)-directions using unit vectors.
The scalar quantities \(r_x\) and \(r_y \) are called the components of \(\vec{r}\text{.}\) When multiplied by the unit vectors \(\hat{x}\) and \(\hat{y}\) they form a vector quantity. The scalar quantities \(r_x\) and \(r_y \) tell us how far we must move in the \(x\)- and \(y\)-direction to be at position \(\vec{r}\text{.}\)
The magnitude, \(r\text{,}\) of vector \(\vec{r}\) (magnitudes will be written without the vector arrows overhead) is the hypotenuse of a right triangle with legs of length \(r_x\) and \(r_y \text{.}\)
\begin{equation*}
r = \sqrt{r_x^2 + r_y^2}
\end{equation*}
Magnitudes are also sometimes written symbolically using bars \(|\vec{r}|\) around the vector. Both representations are equivalent.
ExercisesPractice Activities
1.
Sketch four different vectors so that each vector points in a different quadrant 1 . For each vector, choose an angle and sketch a right triangle whose legs are the components of the vector, like the one in the figure above.
ReferencesReferences
[1]
Vector Decomposition into components by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
[2]
Magnitude and Direction by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
the vector shown above is in the first quadrant; the second quadrant has a negative \(x\)-component; the third quadrant has both components negative; and the fourth quadrant has negative \(y\)-component