### Vector Addition and Geometric Interpretation**Problem Statement:**1. Find the sum of the vectors $\langle3, -1\rangle$ and $\langle-1, 5\rangle$ and illustrate geometrically.**Solution:**To find the sum of the vectors, we add the corresponding components of the given vectors:Let $\mathbf{v_1} = \langle 3, -1 \rangle$ and $\mathbf{v_2} = \langle -1, 5 \rangle$.The sum $\mathbf{v_1} + \mathbf{v_2}$ is:\[\mathbf{v_1} + \mathbf{v_2} = \langle 3, -1 \rangle + \langle -1, 5 \rangle = \langle 3 + (-1), -1 + 5 \rangle = \langle 2, 4 \rangle\]**Geometric Illustration:**To illustrate the sum of these vectors geometrically:1. Plot the initial point (usually the origin) and the terminal points of the vectors $\mathbf{v_1}$ and $\mathbf{v_2}$.2. Since vectors have direction and magnitude but not a fixed position, place the beginning (tail) of $\mathbf{v_2}$ at the end (head) of $\mathbf{v_1}$.3. The resulting vector $\mathbf{v_1} + \mathbf{v_2}$ starts at the initial point (origin) and ends at the terminal point after both vectors are added head-to-tail.On a Cartesian plane:- Draw the vector $\mathbf{v_1} = \langle3, -1\rangle$, starting from the origin (0,0) to the point (3,-1).- From the point (3, -1), draw $\mathbf{v_2} = \langle-1, 5\rangle$, ending at (3 + -1, -1 + 5) = (2, 4).- The vector $\mathbf{v_1} + \mathbf{v_2}$ is a straight line from the origin (0,0) to (2,4).This graphical

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Answered: 1. Find the sum of the vectors (3,-1)…

Math

Calculus

1. Find the sum of the vectors (3,-1) and (-1,5) and illustrate geometrically.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.

Question

### Vector Addition and Geometric Interpretation

**Problem Statement:**
1. Find the sum of the vectors $\langle3, -1\rangle$ and $\langle-1, 5\rangle$ and illustrate geometrically.

**Solution:**
To find the sum of the vectors, we add the corresponding components of the given vectors:

Let $\mathbf{v_1} = \langle 3, -1 \rangle$ and $\mathbf{v_2} = \langle -1, 5 \rangle$.

The sum $\mathbf{v_1} + \mathbf{v_2}$ is:
\[
\mathbf{v_1} + \mathbf{v_2} = \langle 3, -1 \rangle + \langle -1, 5 \rangle
= \langle 3 + (-1), -1 + 5 \rangle
= \langle 2, 4 \rangle
\]

**Geometric Illustration:**
To illustrate the sum of these vectors geometrically:

1. Plot the initial point (usually the origin) and the terminal points of the vectors $\mathbf{v_1}$ and $\mathbf{v_2}$.
2. Since vectors have direction and magnitude but not a fixed position, place the beginning (tail) of $\mathbf{v_2}$ at the end (head) of $\mathbf{v_1}$.
3. The resulting vector $\mathbf{v_1} + \mathbf{v_2}$ starts at the initial point (origin) and ends at the terminal point after both vectors are added head-to-tail.

On a Cartesian plane:
- Draw the vector $\mathbf{v_1} = \langle3, -1\rangle$, starting from the origin (0,0) to the point (3,-1).
- From the point (3, -1), draw $\mathbf{v_2} = \langle-1, 5\rangle$, ending at (3 + -1, -1 + 5) = (2, 4).
- The vector $\mathbf{v_1} + \mathbf{v_2}$ is a straight line from the origin (0,0) to (2,4).

This graphical

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