/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Use a directed line segment to r... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 6

Use a directed line segment to represent the vector $$\mathbf{v}=(-2,-5)$$

Short Answer

Expert verified
The vector \(\mathbf{v}=(-2,-5)\) can be represented as a directed line segment starting from the origin and ending at point (-2, -5) on a coordinate plane.

Step by step solution

01

Identify the components of the vector

The vector given is \(\mathbf{v}=(-2,-5)\). In this 2D vector, \(-2\) is the x-component, and \(-5\) is the y-component. This means the vector will stretch 2 units in the negative x direction and 5 units in the negative y direction from the origin.
02

Draw the x and y-axes

Draw two lines perpendicular to each other. One will be the x-axis and the other the y-axis. Remember that on a standard graph, the x-axis points to the right for positive numbers and, to the left for negative numbers. The y-axis points up for positive numbers and down for negative numbers.
03

Plot the vector

Start from the origin or point (0,0) and move 2 units to the left along the x-axis (since x is -2) then move 5 units down along the y-axis (since y is -5). The ending point is (-2, -5). Draw an arrow starting at the origin to the point (-2, -5). The arrow symbolizes the direction of the given vector.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Vector Components
A vector in a two-dimensional space, like our vector \(\mathbf{v}=(-2,-5)\), is defined by its components. These components are essentially the building blocks of the vector, describing its direction and magnitude. A vector such as \(\mathbf{v}\) has two components: an x-component and a y-component. Here:
  • The x-component is \(-2\). This shows the vector's horizontal movement, which in this case is 2 units to the left since it is negative.
  • The y-component is \(-5\). This indicates the vector's vertical movement, which in this case is 5 units downward due to its negative sign.
To understand vectors better, think of them as arrows pointing from a starting point, often the origin, to a specific location in space. The x and y components direct how far and in which directions the arrow points.
Coordinate System
To plot vectors like \(-2,-5\) on a graph, we use a coordinate system. This system consists of two perpendicular axes: the x-axis and the y-axis. In a typical Cartesian coordinate system:
  • The x-axis runs horizontally, with positive numbers extending to the right and negative numbers to the left.
  • The y-axis runs vertically, with positive values going upwards and negative values going downwards.
The intersection of these two axes is the origin, denoted by the point \((0,0)\). Each coordinate \((x,y)\) represents a unique position on the graph. To plot our vector, we begin at the origin and use the given components to find the final point. The cooperation of x and y coordinates allows us to locate any point on the plane, making it a powerful tool for representing vectors.
Graphical Representation
Graphically representing vectors is a practical way to understand their properties. To visualize \(\mathbf{v}=(-2,-5)\), start by plotting the origin \((0,0)\). From the origin, move according to the vector's components:
  • Shift 2 units to the left due to the x-component \(-2\).
  • Move 5 units downward because of the y-component \(-5\).
Mark the endpoint \((-2,-5)\). Next, draw an arrow from the origin to this point. The arrow's direction shows the vector's direction, and the distance from the origin to the endpoint illustrates its magnitude. This visual makes it easier to interpret how a vector operates in a space and can help solve problems involving direction and displacement effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that a rotation of \(\theta,\) where cot \(2 \theta=(a-c) / b,\) will eliminate the \(x y\) -term from the equation $$a x^{2}+b x y+c y^{2}+d x+e y+f=0$$

Identify and sketch the graph. $$4 x^{2}-y^{2}+4 x+2 y-1=0$$

Test the given set of solutions for linear independence. $$\begin{array}{lll} \text { Differential Equation } & \text { Solutions } \\ y^{\prime \prime \prime \prime}+y^{\prime}=0 & \\{2,-1+2 \sin x, 1+\sin x\\} \end{array}$$

Perform a rotation of axes to eliminate the \(x y\) -term, and sketch the graph of the conic. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}-16=0$$

Use a graphing utility with matrix capabilitics to (a) find the transition matrix from \(B\) to \(B^{\prime},\) (b) find the transition matrix from \(B^{\prime}\) to \(B\), (c) verify that the two transition matrices are inverses of one another, and (d) find [x] \(_{B}\) when provided with \([\mathbf{x}]_{B^{*}}\) $$\begin{array}{l} B=\\{(1,3,4),(2,-5,2),(-4,2,-6)\\} \\\ B^{\prime}=\\{(1,2,-2),(4,1,-4),(-2,5,8)\\} \\\ {[\mathbf{x}]_{B^{\prime}}=\left[\begin{array}{r} -1 \\ 0 \\ 2 \end{array}\right]} \end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.