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Magazine Chapter 17: Problem 9
Find the vector A having \(\overrightarrow{P Q}\) as a representation. Draw \(\overrightarrow{P Q}\) and the position representation of \(A\). $$ P=(-3,5) ; Q=(-5,-2) $$
Short Answer
Expert verified
\[ \overrightarrow{PQ} = (-2, -7) \]
Step by step solution
01
- Understand the Problem
The task is to find the vector \( \overrightarrow{PQ} \) using the points P and Q. Points P and Q are given as P = (-3, 5) and Q = (-5, -2).
02
- Calculate the Components of the Vector
The vector \( \overrightarrow{PQ} \) can be found by subtracting the coordinates of point P from the coordinates of point Q. The formula for this is: \( \overrightarrow{PQ} = Q - P \).\( Q = (-5, -2) \)\( P = (-3, 5) \)
03
- Subtract the Coordinates
Find the difference in the x-coordinates and the y-coordinates separately.For the x-coordinates:\( x_Q - x_P = -5 - (-3) = -5 + 3 = -2 \)For the y-coordinates:\( y_Q - y_P = -2 - 5 = -7 \)
04
- Form the Vector
Combine the differences found in Step 3 to form the vector.\( \overrightarrow{PQ} = (-2, -7) \)
05
- Draw the Vector
Draw points P and Q on a coordinate plane. Then draw an arrow from point P to point Q to represent the vector \( \overrightarrow{PQ} \). Note the direction and the magnitude of the vector.
06
- Position Representation
Show the position vector representation of \( A \), which is the same as \( \overrightarrow{PQ} \), originating from the origin (0, 0) to the point (-2, -7).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
vector subtraction
Vector subtraction is a fundamental concept in coordinate geometry. It allows us to find the displacement between two points in space.
To subtract vectors, you take two points, say P and Q, and subtract their respective coordinates.
In this problem, we start by finding the vector from P to Q, denoted as \( \overrightarrow{PQ} \). This involves subtracting the coordinates of P from those of Q:
This vector gives us a precise description of the direction and distance between the two points P and Q.
To subtract vectors, you take two points, say P and Q, and subtract their respective coordinates.
In this problem, we start by finding the vector from P to Q, denoted as \( \overrightarrow{PQ} \). This involves subtracting the coordinates of P from those of Q:
- For the x-coordinates: \( x_Q - x_P = -5 -(-3) = -5 + 3 = -2 \)
- For the y-coordinates: \( y_Q - y_P = -2 - 5 = -7 \)
This vector gives us a precise description of the direction and distance between the two points P and Q.
coordinate geometry
Coordinate geometry, also known as analytic geometry, deals with the representation of geometric figures using coordinate systems.
This method simplifies calculations and provides a visual understanding of geometric concepts.
In this exercise, we use the Cartesian coordinate system, where each point is defined by a pair of x and y coordinates.
This method simplifies calculations and provides a visual understanding of geometric concepts.
In this exercise, we use the Cartesian coordinate system, where each point is defined by a pair of x and y coordinates.
- First, plot the points P and Q on the coordinate plane: P = (-3, 5) and Q = (-5, -2).
- We then use the aforementioned vector subtraction method to find the displacement vector between these two points.
position vectors
Position vectors are vectors that originate from the origin (0,0) to a specific point in the coordinate system.
They help in locating the positions of points more dynamically.
In our problem, we convert the vector \( \overrightarrow{PQ} \) into a position vector of A.
They help in locating the positions of points more dynamically.
In our problem, we convert the vector \( \overrightarrow{PQ} \) into a position vector of A.
- The vector \( \overrightarrow{PQ} \) has components (-2, -7).
- So, the position vector of A starts from the origin and ends at the point (-2, -7).
vector components
Vectors can be broken down into their components along the coordinate axes. These components are just the differences in the x and y coordinates, making manipulation easy.
Given points P = (-3, 5) and Q = (-5, -2), we already used vector subtraction to find \( \overrightarrow{PQ} \), which gives us the vector components: (-2, -7).
Given points P = (-3, 5) and Q = (-5, -2), we already used vector subtraction to find \( \overrightarrow{PQ} \), which gives us the vector components: (-2, -7).
- The x-component is -2, indicating a leftward movement.
- The y-component is -7, indicating a downward shift.
