Chapter 12: Problem 43
Define the points \(P(-4,1), Q(3,-4),\) and \(R(2,6) .\) Carry out the following calculations. Express \(\overrightarrow{Q R}\) in the form \(a \mathbf{i}+b \mathbf{j}\).
Short Answer
Expert verified
Answer: \(\overrightarrow{QR} = (-1)\mathbf{i} + (10)\mathbf{j}\)
Step by step solution
01
Identify the coordinates of points Q and R
The coordinates of points Q and R are given as \(Q(3, -4)\) and \(R(2, 6)\).
02
Find the difference between the x-components and the y-components of points Q and R
To find the difference between the x-components of points Q and R, subtract the x-coordinate of point Q from the x-coordinate of point R: \(2 - 3 = -1\). Similarly, subtract the y-coordinate of point Q from the y-coordinate of point R: \(6 - (-4) = 6 + 4 = 10\).
03
Express the vector \(\overrightarrow{QR}\) as a linear combination of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\)
We found that the difference between the x-components of points Q and R is -1, and the difference between the y-components is 10. So, we can express the vector \(\overrightarrow{QR}\) in the form \(a\mathbf{i} + b\mathbf{j}\) as \((-1)\mathbf{i} + (10)\mathbf{j}\)
Therefore, the vector \(\overrightarrow{QR}\) in the form \(a\mathbf{i} + b\mathbf{j}\) is \(\boxed{(-1)\mathbf{i} + (10)\mathbf{j}}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Geometry
Coordinate geometry is a crucial branch of mathematics that examines geometry through the use of a coordinate system. It allows us to represent points in a plane using ordered pairs, typically written as \((x, y)\). Each point on a plane is uniquely defined by its x-coordinate and y-coordinate. In the problem, the points given are \((P(-4,1)), (Q(3,-4)), (R(2,6))\).
This system helps in calculating distances between points, finding midpoints, and even describing vectors. When identifying the coordinates of points Q and R, knowing these coordinates enables us to calculate differences in their x and y values. The vector from point Q to point R uses this difference and gives us the foundation forms of mathematical manipulation in coordinate geometry.
Coordinate geometry not only supports basic calculations but also is essential in advanced mathematics, physics, engineering, and computer graphics. It forms the base for creating shapes, analyzing paths, and understanding space dimensions.
This system helps in calculating distances between points, finding midpoints, and even describing vectors. When identifying the coordinates of points Q and R, knowing these coordinates enables us to calculate differences in their x and y values. The vector from point Q to point R uses this difference and gives us the foundation forms of mathematical manipulation in coordinate geometry.
Coordinate geometry not only supports basic calculations but also is essential in advanced mathematics, physics, engineering, and computer graphics. It forms the base for creating shapes, analyzing paths, and understanding space dimensions.
Linear Combination
In mathematics, a linear combination involves multiplying vectors by scalars and then adding the results to produce another vector. This concept applies to expressing vectors in terms of a basis set, usually denoted as unit vectors such as \(\mathbf{i}\) and \(\mathbf{j}\) in two-dimensional space. It allows us to express any vector as a multiplication of these unit vectors with suitable coefficients.
For the vector \((\overrightarrow{QR})\) between points Q and R, we determine the coefficients by subtracting the coordinates of Q from R, giving us the values used for our linear combination. Simply put, a vector can be written as \(a\mathbf{i} + b\mathbf{j}\), where \((a, b)\) are the coefficients coming from the differences in the respective \((x, y)\) coordinates.
Understanding linear combinations is essential for expressing vector relationships and is a building block for more complex topics in vector spaces and linear algebra. It allows for flexibility and manipulation of vector quantities, useful in multiple fields like physics, where forces or velocities need representation in vector forms.
For the vector \((\overrightarrow{QR})\) between points Q and R, we determine the coefficients by subtracting the coordinates of Q from R, giving us the values used for our linear combination. Simply put, a vector can be written as \(a\mathbf{i} + b\mathbf{j}\), where \((a, b)\) are the coefficients coming from the differences in the respective \((x, y)\) coordinates.
Understanding linear combinations is essential for expressing vector relationships and is a building block for more complex topics in vector spaces and linear algebra. It allows for flexibility and manipulation of vector quantities, useful in multiple fields like physics, where forces or velocities need representation in vector forms.
Unit Vectors
Unit vectors are vectors that have a magnitude of 1. They serve as the building blocks for other vectors, allowing for straightforward representation and manipulation within a vector space. In two-dimensional space, the unit vectors are typically \(\mathbf{i}\) and \(\mathbf{j}\), representing the horizontal and vertical components, respectively.
Unit vectors are critical in defining any vector as a linear combination of these fundamental components. By assignment of coefficients to \(\mathbf{i}\) and \(\mathbf{j}\), we can express any vector within a plane, such as \(\overrightarrow{QR} = -1\mathbf{i} + 10\mathbf{j}\). This form offers a simplified notation to express how a vector stretches across dimensions.
The efficiency of unit vectors lies in their ability to simplify complex vector calculations and enhance the understanding of multidimensional spaces. Engineers, physicists, and computer scientists widely use unit vectors for tasks ranging from designing complex architectures to simulating physical phenomena and even in graphics rendering. Their simplicity and versatility make them indispensable in the scientific arena.
Unit vectors are critical in defining any vector as a linear combination of these fundamental components. By assignment of coefficients to \(\mathbf{i}\) and \(\mathbf{j}\), we can express any vector within a plane, such as \(\overrightarrow{QR} = -1\mathbf{i} + 10\mathbf{j}\). This form offers a simplified notation to express how a vector stretches across dimensions.
The efficiency of unit vectors lies in their ability to simplify complex vector calculations and enhance the understanding of multidimensional spaces. Engineers, physicists, and computer scientists widely use unit vectors for tasks ranging from designing complex architectures to simulating physical phenomena and even in graphics rendering. Their simplicity and versatility make them indispensable in the scientific arena.