/*! 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 11 Give the vector equation of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 11

Give the vector equation of the line passing through \(P\) and \(Q\). $$P=(1,-2), Q=(3,0)$$

Short Answer

Expert verified
The vector equation is \( \mathbf{r}(t) = (1, -2) + t(2, 2) \).

Step by step solution

01

Find the Direction Vector

To find the direction vector of the line, subtract the coordinates of point \(P\) from point \(Q\). The direction vector \( \mathbf{d} \) is calculated as: \[ \mathbf{d} = (3 - 1, 0 - (-2)) = (2, 2) \]
02

Write the Vector Equation

The vector equation of a line can be written using a point on the line and the direction vector. Here, we can use point \(P = (1, -2)\) and the direction vector \(\mathbf{d} = (2, 2)\). The vector equation is given by:\[ \mathbf{r}(t) = (1, -2) + t(2, 2) \]where \( t \) is a parameter.

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.

Direction Vector
When you want to describe a line in coordinate geometry, the direction vector is a crucial concept. It defines the line's orientation and indicates the "direction" the line is heading. To find the direction vector of a line through two points, such as point \( P = (1, -2) \) and point \( Q = (3, 0) \), you simply perform subtraction:
  • Subtract the coordinates of the starting point from the ending point.
  • This gives you the vector \( \mathbf{d} = (3 - 1, 0 - (-2)) = (2, 2) \).

Think of the direction vector like an arrow: it starts at the first point and ends at the second. No matter where you place this arrow on the coordinate plane, it will always point in the same direction.
Parametric Form
The parametric form of a line is a way to express the line's equation using parameters. This method uses a vector equation which is often more flexible than a standard linear equation. It is especially useful in physics and engineering.To express the line in parametric form, you need:
  • A point on the line, like \( P = (1, -2) \).
  • The direction vector, which we've calculated as \( \mathbf{d} = (2, 2) \).
Combining these, the vector equation is:\[ \mathbf{r}(t) = (1, -2) + t(2, 2) \]Here, \( t \) is a parameter that can be any real number. By changing \( t \), you can move back and forth along the line, effectively tracing out every possible point on the line.
Points in 2D
Points in two-dimensional space, or 2D, are fundamental elements in coordinate geometry. Each point is defined by two numbers or coordinates, such as \( P = (1, -2) \) and \( Q = (3, 0) \), which tell you the position on the Cartesian plane.Using these points, we can visualize lines, planes, and curves. Points serve as the building blocks for more complex geometric shapes. By connecting two points with a line, we apply the concept of vectors to navigate between them in 2D space.
  • Each coordinate pair consists of an \( x \)-value and a \( y \)-value.
  • The \( x \)-value represents the horizontal position.
  • The \( y \)-value represents the vertical position.

Understanding points in 2D helps in graphing and solving geometric problems effectively.
Coordinate Geometry
Coordinate geometry, sometimes known as analytic geometry, blends algebra and geometry to investigate geometric shapes and figures using coordinates and equations. Think of it as a bridge between algebraic formulas and geometric figures.By using a coordinate system, like a Cartesian plane, we can describe any geometric shape by an equation or a set of equations.
  • Lines, like the one through \( P \) and \( Q \), can be expressed as vector equations or in standard form.
  • It simplifies measuring distances and angles.
  • It facilitates the finding of intersection points and other properties of geometric figures.

With coordinate geometry, solving geometric problems becomes more manageable, as it provides tools for both visualization and calculation.

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

Using mathematical induction, prove the following generalization of the Triangle Inequality: $$\begin{array}{l} \left\|\mathbf{v}_{1}+\mathbf{v}_{2}+\cdots+\mathbf{v}_{n}\right\| \leq\left\|\mathbf{v}_{1}\right\|+\left\|\mathbf{v}_{2}\right\|+\cdots+\left\|\mathbf{v}_{n}\right\| \\\ \text { for all } n \geq 1 \end{array}$$

Perform the indicated calculations. $$[2,1,2]+[2,0,1] \text { in } \mathbb{Z}_{3}^{3}$$

Determine whether the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is acute, obtuse, or a right angle. $$\mathbf{u}=\left[\begin{array}{r} 2 \\ -1 \\ 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \\ -1 \end{array}\right]$$

(a) For which values of \(a\) does \(a x=1\) have a solution in \(\mathbb{Z}_{5} ?\) (b) For which values of \(a\) does \(a x=1\) have a solution in \(\mathbb{Z}_{6} ?\) (c) For which values of \(a\) and \(m\) does \(a x=1\) have a solution in \(\mathbb{Z}_{m} ?\)

Show that the plane and line with the given equations intersect, and then find the acute angle of intersection between them. The plane given by \(x+y+2 z=0\) and the line \\[ \begin{aligned} \operatorname{given} \operatorname{by} x &=2+t \\ y &=1-2 t \\ z &=3+t \end{aligned} \\]
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.