Problem 4 - Calculus: Early Transcendentals

Question

Let \(v\) be the vector with tail at the origin and head at (-1,2)\(;\) let \(w\) be the vector with tail at the origin and head at \((3,3) .\) Draw \(v\) and \(w\) and a vector \(u\) with tail at (-1,2) and head at (3,3) . Draw \(\boldsymbol{u}\) with its tail at the origin.

Step-by-Step Solution

Answer

Vector \(u\) is \(\begin{bmatrix} 4 \\ 1 \end{bmatrix}\) and runs from the origin to (4,1).

Step 1: Understand the Given Vectors

Vector \(v\) has its tail at the origin and head at (-1, 2). Therefore, vector \(v\) can be represented as \(\begin{bmatrix} -1 \ 2 \end{bmatrix}\). Similarly, vector \(w\) has its tail at the origin and head at (3, 3), represented as \(\begin{bmatrix} 3 \ 3 \end{bmatrix}\).

Step 2: Determine the Vector \(u\)

The vector \(u\) starts from the head of vector \(v\) and ends at the head of vector \(w\). To determine \(u\), subtract the coordinates of the head of \(v\) from the coordinates of the head of \(w\):\[ u = \begin{bmatrix} 3 - (-1) \ 3 - 2 \end{bmatrix} = \begin{bmatrix} 4 \ 1 \end{bmatrix} \].

Step 3: Draw Vectors v, w, and u

Plot vector \(v\) from the origin (0,0) to (-1,2). Plot vector \(w\) from the origin to (3,3). Next, plot vector \(u\) from (-1,2) to (3,3). The vector \(u\) should start at (-1,2) and extend to (3,3), following the previously calculated components.

Step 4: Draw Vector \(u\) with Tail at Origin

To draw \(u\) with its tail at the origin, maintain the same direction and magnitude as before but start it at the origin. With \(u = \begin{bmatrix} 4 \ 1 \end{bmatrix}\), visualize vector \(u\) from (0,0) to (4,1).

Key Concepts

Vector Addition

Vector addition is the process of combining vectors to form a resultant vector. Consider vectors as arrows. Each has a direction and magnitude. To add vectors, you position the tail of one vector to the head of another. This is also known as the "tip-to-tail" method. In our problem, vectors \( v \) and \( u \) are combined to achieve vector \( w \).

• Vector \( v \) is given by \( \begin{bmatrix} -1 \ 2 \end{bmatrix} \), and vector \( u \) is \( \begin{bmatrix} 4 \ 1 \end{bmatrix} \).
• When added: \( \begin{bmatrix} -1 \ 2 \end{bmatrix} + \begin{bmatrix} 4 \ 1 \end{bmatrix} = \begin{bmatrix} 3 \ 3 \end{bmatrix} \).
• The result is vector \( w \) from the origin to the point (3,3).

Visualize this as moving in the direction of \( v \) and then in the direction of \( u \) to reach the end of \( w \). In vector terms, it confirms \( v + u = w \).

Vector Representation

Vectors are represented both geometrically and algebraically. Geometrically, they are shown as arrows with specific directions and lengths. Algebraically, they are described using coordinates based on their heads and tails in a two-dimensional space.

• Vector \( v \) extends from the origin (0,0) to (-1,2).
• This makes vector \( v \) synonymous with the point \( \begin{bmatrix} -1 \ 2 \end{bmatrix} \).
• Similarly, vector \( w \) extends from the origin to (3,3) represented by \( \begin{bmatrix} 3 \ 3 \end{bmatrix} \).

The representation system allows easy calculations such as addition or finding distances. Having clear representations allows vectors to be manipulated in a straightforward manner, crucial for effectively solving problems in vector calculus. To represent vector \( u \), observe that \( u \) starts from the head of \( v \) and ends at the head of \( w \), represented as \( \begin{bmatrix} 4 \ 1 \end{bmatrix} \).

Coordinate Systems

Coordinate systems are structures that define the space we're working in. They allow us to locate points and vectors using ordered pairs. In a two-dimensional plane, these pairs are typically written as (x, y).

• The Cartesian coordinate system is often used, where vectors are laid out on a grid with horizontal (x-axis) and vertical (y-axis) dimensions.
• Vector \( v \) is at (-1,2), meaning it moves -1 step on the x-axis and 2 steps on the y-axis from the origin.
• Vector \( w \) and vector \( u \) adhere to similar principles, using coordinates (3,3) and (4,1), respectively.

Coordinate systems enable us to perform operations like addition and subtraction by aligning vectors in space. They serve as the backbone of vector calculus, helping visualize and compute vector properties easily. Understanding these settings simplifies navigating vectors' mathematical operations and drawing diagrams as required by the problem at hand.