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Chapter 16: Problem 4

Find a single vector resulting from the operations. \((x, y, z)+(2,6,8)\)

Short Answer

Expert verified
Answer: The resulting single vector is \((x + 2, y + 6, z + 8)\).

Step by step solution

01

Add the x-components

To add the x-components of the two given vectors, we just need to add their corresponding x-coordinates. In this case, our x-coordinates are \(x\) and \(2\). So, we add them together: \(x + 2\).
02

Add the y-components

Now, we will add the y-components of the two given vectors. The y-coefficients are \(y\) and \(6\). We will add them together: \(y + 6\).
03

Add the z-components

Finally, we will add the z-components of the two given vectors. The z-coordinates are \(z\) and \(8\). We will add them: \(z + 8\).
04

Combine the results

Now that we have added the x, y, and z-components of the two given vectors, we just need to combine the results into a single vector. The resulting vector will have the form \((x + 2, y + 6, z + 8)\). So, the single vector resulting from the operations is \((x + 2, y + 6, z + 8)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
Vectors are essential in understanding many physical and mathematical phenomena. A vector is typically represented by its components along the coordinate axes. Each vector component shows the magnitude of the vector in one of the directions of a coordinate system. For example:
  • The x-component measures how far the vector extends in the x-direction.
  • The y-component measures the extent in the y-direction.
  • The z-component measures the reach in the z-direction.
These components are crucial in performing operations like addition because they provide a way to systematically combine vectors. When adding vectors, it's important to sum up each of the respective components to form a new resultant vector. This process ensures the final vector accurately represents the combined influence of the original vectors in each direction.
Three-Dimensional Vectors
Three-dimensional (3D) vectors are used to describe points or directions in a space that has three dimensions, like our physical world. They are written as \(x, y, z\), where:
  • \(x\) is the position along the x-axis.
  • \(y\) is the position along the y-axis.
  • \(z\) is the position along the z-axis.
Each component corresponds to one dimension in the space, so 3D vectors can fully describe movement or position in the entirety of three-dimensional space. They are quite helpful in complex fields like physics and engineering, where objects often move in all three dimensions. When dealing with 3D vectors, it's important to process each component individually, as they each fully describe the vector in their respective dimensions.
Coordinate System
A coordinate system allows you to precisely describe the location of a point or the direction and magnitude of a vector. In a three-dimensional space, you typically use the Cartesian coordinate system characterized by:
  • The x-axis running horizontally.
  • The y-axis running vertically (in a plane perpendicular to the x-axis).
  • The z-axis running vertically (in another plane, perpendicular to both the x- and y-axes).
Each axis forms a right-angle intersection with the others, creating an orthogonal system that uniquely defines any position in space. This orthogonality is beneficial because it makes mathematical operations simpler and less error-prone. By using this system, you can easily visualize and manipulate vectors in three dimensions, ensuring clarity and precision in your calculations.

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Most popular questions from this chapter

Find a single vector resulting from the operations. \((5,7,13)+(2,6,8)\)

What do you need to know about two matrices to know if their product exists?$$ \begin{aligned} &\text { Given } \mathbf{G}=\left(\begin{array}{ll} 5 & 2 \\ 1 & 3 \end{array}\right) \text { and } \mathbf{H}=\left(\begin{array}{cc} -2 & 4 \\ 3 & -1 \end{array}\right), \text { verify the }\\\ &\text { statements } \text { . } \end{aligned} $$ $$ 3(\mathbf{G} \mathbf{H})=(3 \mathbf{G}) \mathbf{H} $$

If possible, use row operations to solve the systems. $$ \left\\{\begin{array}{l} x+y=5 \\ 2 x-y=7 \end{array}\right. $$

Check the statements in Exercises \(19-23\) using the matrices \(\mathbf{U}=\left(\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right), \mathbf{V}=\left(\begin{array}{cc}-1 & 4 \\ 0 & 2\end{array}\right), \mathbf{W}=\left(\begin{array}{cc}5 & -5 \\ 4 & 7\end{array}\right).\) $$ 2(3 \mathbf{V})=6 \mathbf{V} $$

Find a single vector resulting from the operations. \(\left(\begin{array}{cc}5 & 10 \\ 20 & 40\end{array}\right)\left(\begin{array}{l}3 \\ 7\end{array}\right)\)
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