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Chapter 12: Problem 8

Given two points \(P\) and \(Q,\) how are the components of \(\overrightarrow{P Q}\) determined?

Short Answer

Expert verified
Question: Determine the components of the vector $\overrightarrow{P Q}$, given the coordinates of point $P(3, 4)$ and point $Q(8, 6).$ Answer: To find the components of the vector $\overrightarrow{P Q}$, first subtract the x-coordinates: $x_{\overrightarrow{P Q}} = 8 - 3 = 5$. Then subtract the y-coordinates: $y_{\overrightarrow{P Q}} = 6 - 4 = 2$. Therefore, $\overrightarrow{P Q} = \langle 5, 2 \rangle$.

Step by step solution

01

Identify the Coordinates of Both Points

Each point has an \(x\) and \(y\) coordinate in a 2-dimensional Cartesian coordinate system. Let's denote the coordinates of point \(P\) as \((x_{1}, y_{1})\) and the coordinates of point \(Q\) as \((x_{2}, y_{2})\).
02

Find the Difference in \(x\)- and \(y\)-coordinates

Subtract the \(x\)-coordinate of point \(P\) from the \(x\)-coordinate of point \(Q\) to find the \(x\)-component of \(\overrightarrow{P Q}\). Similarly, subtract the \(y\)-coordinate of point \(P\) from the \(y\)-coordinate of point \(Q\) to find the \(y\)-component of \(\overrightarrow{P Q}\). The equations will be as follows: - \(x_{\overrightarrow{P Q}} = x_{2} - x_{1}\) - \(y_{\overrightarrow{P Q}} = y_{2} - y_{1}\)
03

Write the Components of \(\overrightarrow{P Q}\) as a Vector

Now, write the components as a vector using the convention for representing vectors: \(\overrightarrow{P Q} = \langle x_{\overrightarrow{P Q}}, y_{\overrightarrow{P Q}}\rangle\) So, the components of the vector \(\overrightarrow{P Q}\) given the coordinates of points \(P\) and \(Q\) can be determined using the above steps.

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