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Subsection 2.8.7 Position Vectors
Activity 2.8.7 .
What you require: Graph paper
(a) Draw the
\(x\) and
\(y\) axis on the graph paper as shown below.
(b) Plot the following points
\(A(1,1),B(3,5),C(2,1),D(4,-3) \) on the graph.
(c) Draw a directed line from
\(A\) to
\(B\) to represent
\(\overrightarrow{AB}\text{.}\)
(d) Draw another directed line from
\(D\) to
\(C\) to represent
\(\overrightarrow{CD}\text{.}\)
(e) Determine the position vector of
\(B\) relative to point
\(A\text{.}\)
(f) Determine the position vector of
\(C\) relative to point
\(D\text{.}\)
(g) Discuss and share your findings with the rest of the class.
\(\textbf{Key Takeaway}\)
In
Figure 2.8.38 below, points
\(A(2,3)\) and
\(B(5,1)\) are located in the plane relative to origin point O in the plane.
The position vector of
\(A\) is
\(\textbf{OA} = \begin{pmatrix} 2 - 0 \\ 3 - 0 \end{pmatrix} = \binom{2}{3}\)
The position vector of
\(B\) is
\(\textbf{OB} = \begin{pmatrix} 5 - 0 \\ 1 - 0 \end{pmatrix} = \binom{5}{1}\)
Similarly, for point
\(A\) in the plane its position vector
\(\textbf{OA}\) is denoted by
\(\mathbf{a}\text{.}\) Also for point
\(B\) in the plane it’s position vector
\(\textbf{OB}\) is denoted by
\(\mathbf{b}\text{.}\)
Figure 2.8.38.
Example 2.8.39 .
Find the position vector of
\(A\) and
\(B\text{.}\)
Solution .
The position vector of A is
\(\textbf{OA } = \begin{pmatrix} 6 - 0 \\ -2 - 0\end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}\)
Similarly the position vector of B is
\(\textbf{OB }= \begin{pmatrix} 4 - 0 \\ -4 - 0\end{pmatrix} = \begin{pmatrix} 4 \\ -4 \end{pmatrix}\)
Exercises Exercises
1. Draw the following position vector on a graph paper:
\(a = \left( \begin{matrix} 10 \\ -2 \end{matrix}\right)\)
\(b = \left( \begin{matrix} 2 \\ 5 \end{matrix}\right)\)
\(c = \left( \begin{matrix} -3 \\ 8 \end{matrix}\right)\)
\(d = \left( \begin{matrix} -11 \\ 6 \end{matrix}\right)\)
\(e = \left( \begin{matrix} 4 \\ 4 \end{matrix}\right)\)
\(f = \left( \begin{matrix} -1 \\ 12 \end{matrix}\right)\)
\(g = \left( \begin{matrix} 5 \\ 3 \end{matrix}\right)\)
\(h = \left( \begin{matrix} 0 \\ 13 \end{matrix}\right)\)
2. Use
Figure 2.8.40 below to write the position vectors of points
\(M,E,D,A\text{.}\)
Figure 2.8.40.
Checkpoint 2.8.41 .
