Problem 12
Question
In Exercises \(9-16,\) find the component form of the vector. The sum of \(\overrightarrow{A B}\) and \(\overrightarrow{C D},\) where \(A=(1,-1), B=(2,0)\) \(C=(-1,3),\) and \(D=(-2,2)\)
Step-by-Step Solution
Verified Answer
The component form of \( \overrightarrow{A B} + \overrightarrow{C D} \) is \( (0, 0) \).
1Step 1: Identify Components of Vectors
First, identify the components of vectors \( \overrightarrow{A B} \) and \( \overrightarrow{C D} \). To find these, subtract the coordinates of the initial point from the coordinates of the terminal point for each vector.For \( \overrightarrow{A B} \):\( A = (1, -1) \)\( B = (2, 0) \)The components of \( \overrightarrow{A B} \) are:\[\overrightarrow{A B} = (2-1, 0-(-1)) = (1, 1)\]For \( \overrightarrow{C D} \):\( C = (-1, 3) \)\( D = (-2, 2) \)The components of \( \overrightarrow{C D} \) are:\[\overrightarrow{C D} = (-2 - (-1), 2 - 3) = (-1, -1)\]
2Step 2: Add the Components of the Vectors
Next, add the respective components of vectors \( \overrightarrow{A B} \) and \( \overrightarrow{C D} \) to find the component form of their sum.The vectors \( \overrightarrow{A B} = (1, 1) \) and \( \overrightarrow{C D} = (-1, -1) \) are added as follows:\[\overrightarrow{A B} + \overrightarrow{C D} = (1 + (-1), 1 + (-1)) = (0, 0)\]
3Step 3: State the Result
After performing the addition, the result is that the component form of the sum of the vectors \( \overrightarrow{A B} \) and \( \overrightarrow{C D} \) is \( (0, 0) \).
Key Concepts
Component form of vectorsVector componentsVector arithmetic
Component form of vectors
Vectors are mathematical objects used to represent quantities that have both a magnitude and a direction. In mathematics, particularly in vector spaces, these entities can be more easily manipulated when expressed in a specific format: the component form.
When converting a vector into component form, you first need to identify its initial and terminal points, as vectors are defined by points in a coordinate system. For instance, consider vector \( \overrightarrow{A B} \) with initial point \( A = (1, -1) \) and terminal point \( B = (2, 0) \). To place this vector into component form, subtract the coordinates of the initial point from those of the terminal point. This gives us the directed distance in each dimension:
This process converts any vector into an ordered pair that succinctly describes its reach in each axis, making further calculations more straightforward.
When converting a vector into component form, you first need to identify its initial and terminal points, as vectors are defined by points in a coordinate system. For instance, consider vector \( \overrightarrow{A B} \) with initial point \( A = (1, -1) \) and terminal point \( B = (2, 0) \). To place this vector into component form, subtract the coordinates of the initial point from those of the terminal point. This gives us the directed distance in each dimension:
- Horizontal component: \( 2 - 1 = 1 \)
- Vertical component: \( 0 - (-1) = 1 \)
This process converts any vector into an ordered pair that succinctly describes its reach in each axis, making further calculations more straightforward.
Vector components
Every vector can be simplified into what are called its components. These components are essentially the building blocks of the vector, indicating how far, along each axis in the coordinate system, the vector extends.
The concept of vector components is essential in understanding vectors in physics, engineering, and mathematics. Consider vector \( \overrightarrow{C D} \) with initial point \( C = (-1, 3) \) and terminal point \( D = (-2, 2) \). To find the components of this vector, you subtract each coordinate of point \( C \) from point \( D \):
Each vector's components help visualize the vector's effect in two-dimensional space. In essence, every vector is the sum of its parallel movements along each coordinate axis.
The concept of vector components is essential in understanding vectors in physics, engineering, and mathematics. Consider vector \( \overrightarrow{C D} \) with initial point \( C = (-1, 3) \) and terminal point \( D = (-2, 2) \). To find the components of this vector, you subtract each coordinate of point \( C \) from point \( D \):
- Horizontal component: \( -2 - (-1) = -1 \)
- Vertical component: \( 2 - 3 = -1 \)
Each vector's components help visualize the vector's effect in two-dimensional space. In essence, every vector is the sum of its parallel movements along each coordinate axis.
Vector arithmetic
Vector arithmetic involves performing operations such as addition, subtraction, and scalar multiplication on vectors. These operations are crucial in various fields for combining directions and magnitudes.
Let's delve into the addition of vectors, which is simply the sum of their respective components. For vectors \( \overrightarrow{A B} = (1, 1) \) and \( \overrightarrow{C D} = (-1, -1) \), their addition is straightforward:
This resulting vector \((0, 0)\) indicates equilibrium, where there is no net movement either horizontally or vertically. Understanding vector arithmetic enables you to combine forces, motions, or other vector quantities effectively.
Let's delve into the addition of vectors, which is simply the sum of their respective components. For vectors \( \overrightarrow{A B} = (1, 1) \) and \( \overrightarrow{C D} = (-1, -1) \), their addition is straightforward:
- Sum of horizontal components: \( 1 + (-1) = 0 \)
- Sum of vertical components: \( 1 + (-1) = 0 \)
This resulting vector \((0, 0)\) indicates equilibrium, where there is no net movement either horizontally or vertically. Understanding vector arithmetic enables you to combine forces, motions, or other vector quantities effectively.
Other exercises in this chapter
Problem 12
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In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x^{2}+(y-1)^{2}+z^{
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Sketch the surfaces in Exercises \(13-76\) $$ x^{2}+y^{2}=4 $$
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