1.
Sketch a unit vector that points in the same direction as the vector \(\vec{v_1}= 3\hat{x}-4\hat{y}\text{.}\)
Section 1.4 Unit Vectors
You will often need to orient a vector within a particular coordinate system. A unit vector is a vector that has magnitude of one. For this reason, unit vectors are often called direction vectors.
You represent a unit vector symbolically with a "hat." Most commonly, unit vectors are used for indicating the directions of a coordinate system. For example \(\hat{x} \text{,}\) \(\hat{y} \) and \(\hat{z} \) are unit vectors which point parallel to the positive \(x\)-direction, positive \(y\)-direction and positive \(z\)-direction (these are often called basis vectors in a mathematics class).
Since \(\hat{x} \text{,}\) \(\hat{y} \) and \(\hat{z} \) are mutually perpendicular unit vectors, their dot products
\begin{equation*}
\hat{x} \cdot \hat{x} = \hat{y} \cdot \hat{y} = \hat{z} \cdot \hat{z} = 1
\end{equation*}
\begin{equation*}
\hat{x} \cdot \hat{y} = \hat{x} \cdot \hat{z} = \hat{y} \cdot \hat{z} = 0
\end{equation*}
Many textbooks, and authors use different notation for the unit vectors. Often, you will see unit vectors written using \(\hat{i} \text{,}\) \(\hat{j} \) and \(\hat{k} \) to represent the positive \(x\)-direction, positive \(y\)-direction, and positive \(z\)-direction. Both notations are equivalent, and are just a convention. Use whatever notation is more comfortable for you.
This short video includes a brief summary of unit vectors:
Exercises Practice Activities
2.
Sketch a unit vector that points perpendicular to the vector \(\vec{v_2}= -12\hat{x}-5\hat{y}\text{.}\)
References References
[1]
Coordinate systems and unit vectors by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
