 8.5 Vectors in Cartesian Plane

(A) Vectors in Cartesian Coordinates
1. A unit vector is a vector whose magnitude is one unit.
2. A unit vector that is parallel to the x-axis is denoted by  $\underset{˜}{i}$ while a unit vector that is parallel to the y-axis is denoted by $\underset{˜}{j}$ .
3. The unit vector can be expressed in columnar form as below:
4. The magnitudes of the unit vectors are  $|\underset{˜}{i}|=|\underset{˜}{j}|=1.$

5. The magnitude of the vector $\stackrel{\to }{OA}$ can be calculated using the Pythagoras’ Theorem.
$\overline{)\text{}|\stackrel{\to }{OA}|=\sqrt{{x}^{2}+{y}^{2}}\text{}}$

(B) Unit Vector in the Direction of a Vector

Example 1:
If   $\underset{˜}{r}=k\underset{˜}{i}-8\underset{˜}{j}$ and $|\underset{˜}{r}|=10$ , find the values of k. Determine the unit vector in the direction of   $\underset{˜}{r}$   for each value of k.

Solution:

Example 2:
It is given that
(a) Find
(b) Hence, find the unit vector in the direction of $\underset{˜}{b}-\underset{˜}{a}\text{}.$

Solution:
(a)
$\begin{array}{l}\underset{˜}{b}-\underset{˜}{a}=\left(\begin{array}{l}3\\ 7\end{array}\right)-\left(\begin{array}{l}6\\ 3\end{array}\right)=\left(\begin{array}{l}3-6\\ 7-3\end{array}\right)=\left(\begin{array}{l}-3\\ \text{}4\end{array}\right)\\ \\ |\underset{˜}{b}-\underset{˜}{a}|=\sqrt{{\left(-3\right)}^{2}+{4}^{2}}=\sqrt{9+16}=\sqrt{25}=5\end{array}$

(b)