The i-j System
Consider the following figure, which shows a point P with coordinates \(\left( {3,4} \right)\) on a rectangular coordinate system with origin O:
How can we completely specify the vector \(\overrightarrow {OP} \) in a mathematical sense? Note that to reach P from O, we can first move along the vector \(\overrightarrow {OQ} \), and then along the vector \(\overrightarrow {QP} \). Using the triangle law, we can write the following relation:
\[\overrightarrow {OP} = \overrightarrow {OQ} + \overrightarrow {QP} \]
Now, \(\overrightarrow {OQ} \) is a vector 3 units long, and points in the positive x-direction. Let us denote a unit vector in the positive x-direction by \(\hat i\). Then, we can write \(\overrightarrow {OQ} \) as \(3\hat i\).
Similarly, we observe that \(\overrightarrow {QP} \) is a vector 4 units long, and points in the positive y-direction. If we denote a unit vector in the positive y-direction by \(\hat j\), we can write \(\overrightarrow {QP} \) as \(4\hat j\).
These observations are included in the following figure:
Thus, we can write \(\overrightarrow {OP} \) in terms of \(\hat i\) and \(\hat j\) as follows:
\[\overrightarrow {OP} = \overrightarrow {OQ} + \overrightarrow {QP} = 3\hat i + 4\hat j\]
Now, \(\overrightarrow {OP} \) is completely specified in a mathematical sense, since this representation tells us exactly how to reach P from O: first move 3 units along the positive x-direction ( \(3\hat i\)), and then move 4 units parallel to the positive y-direction ( \(4\hat j\)).
The following figure shows a point A with coordinates \(\left( { - 4,1} \right)\):
In terms of \(\hat i\) and \(\hat j\), we can write the vector \(\overrightarrow {OA} \) as follows:
\[\overrightarrow {OA} = \overrightarrow {OB} + \overrightarrow {BA} = - 4\hat i + \hat j\]
Consider the following figure, which shows a right-angled triangle ABC with the specified coordinates:
Note that vector is 5 units long, and points in the positive x-direction. Thus, we can write \(\overrightarrow {AB} = 5\hat i\). Vector \(\overrightarrow {BC} \) points in the negative y-direction and is 2 units long, so we can write \(\overrightarrow {BC} = - 2\hat j\). Using the triangle law, we have:
\[\overrightarrow {AC} = \overrightarrow {AB} + \overrightarrow {BC} = 4\hat i - 2\hat j\]
Example 1: Observe the following figure, which shows three vectors \(\overrightarrow a \), \(\overrightarrow b \) and \(\overrightarrow c \):
Specify these vectors in terms of \(\hat i\) and \(\hat j\).
Solution: To move from the starting point of vector \(\overrightarrow a \) to its tip, we need to move 4 units in the negative x-direction, and 2 units in the negative y-direction:
Thus,
\[\overrightarrow a = - 4\hat i - 2\hat j\]
Following a similar approach, we have:
\[\begin{array}{l}\overrightarrow b = 2\hat i - 5\hat j\\\overrightarrow c = - 3\hat i + 2\hat j\end{array}\]