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.
Section 1.6 Vector Components
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.
You can write \(\vec{r}\) in terms of its components in the \(x\)- and \(y\)-directions using unit vectors.
\begin{equation*}
\vec{r} = r_x \hat{x} + r_y\hat{y}
\end{equation*}
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.
Exercises Practice Activities
References References
[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
