Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Short Answer

To describe the set of all points for condition \(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k\).

All set of points for condition \(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k\) are described and it represents an ellipse.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept of Two-Dimensional Vector

The magnitude of a vector refers to the entire amount of the quantity it represents. The magnitude of a two-dimensional vector is equal to the length of the hypotenuse of a triangle with the \({\bf{x}}\) - and \(\;{\bf{y}}\) -components as sides.

Given:

Three two-dimensional vectors \({\bf{r}} = \langle x,y\rangle ,{{\bf{r}}_1} = \left\langle {{x_1},{y_1}} \right\rangle \) and \({{\bf{r}}_2} = \left\langle {{x_2},{y_2}} \right\rangle \).

Formula used:

Write the expression for ellipse with center \(C(h,k)\).

\(\frac{{{{(x - h)}^2}}}{{{a^2}}} + \frac{{{{(y - k)}^2}}}{{{b^2}}} = 1.......(1)\)

Here,

\(x,{\rm{ }}y\), and \(z\) are Cartesian coordinates.

Step 2: Calculate sets of all points for condition

Write the expression for relation between vectors \({\bf{r}},{{\bf{r}}_1}\) and \({{\bf{r}}_2}\).

\(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k.......(2)\)

Consider \({{\bf{r}}_1}\) and \({{\bf{r}}_2}\) as points \({P_1}\) and \({P_2}\) respectively. The equation (2) indicates the sum of distances between \((x,y)\) and points \({P_1}\) and \({P_2}\). According to the equation (1) set of points in equation (2) represents foci of ellipse. The no degenerating condition of ellipse is \(k > \left| {{{\bf{r}}_1} - {{\bf{r}}_2}} \right|\).

Thus, all set of points for condition \(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k\) are described and it represents an ellipse.

Find Study Materials for

Create Study Materials

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

To describe the set of all points for condition \(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k\).

Step by Step Solution

Step 1: Concept of Two-Dimensional Vector

Step 2: Calculate sets of all points for condition

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

To describe the set of all points for condition \(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k\).

Step by Step Solution

Step 1: Concept of Two-Dimensional Vector

Step 2: Calculate sets of all points for condition

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths