Problem 29 - Calculus for Biology and Medicine

Question

In Problems 29-34, let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] $$ $$ \text { Compute } \mathbf{u}+\mathbf{v} \text { and illustrate the result graphically. } $$

Step-by-Step Solution

Answer

\( \mathbf{u} + \mathbf{v} = \left[ \begin{array}{c} 2 \\ 2 \end{array} \right] \). Graphically, it is a vector from the origin to (2, 2).

Step 1: Add the Vectors

To find \( \mathbf{u} + \mathbf{v} \), we add the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \). That means, add the first component of \( \mathbf{u} \) to the first component of \( \mathbf{v} \), and then add the second component of \( \mathbf{u} \) to the second component of \( \mathbf{v} \). Mathematically:\[\mathbf{u} + \mathbf{v} = \left[\begin{array}{c} 3 \ 4 \end{array}\right] + \left[\begin{array}{c} -1 \ -2 \end{array}\right] = \left[\begin{array}{c} 3 + (-1) \ 4 + (-2) \end{array}\right] = \left[\begin{array}{c} 2 \ 2 \end{array}\right]\]

Step 2: Graphical Representation

To illustrate the result graphically, plot the vectors \( \mathbf{u} \), \( \mathbf{v} \), and their resultant \( \mathbf{u} + \mathbf{v} \) on an XY plane. Start \( \mathbf{u} \) at the origin and draw a vector to \((3, 4)\). Then, starting at \((3, 4)\), draw the vector \( \mathbf{v} \) falling to \((2, 2)\). The resultant vector \( \mathbf{u} + \mathbf{v} \) connects the origin directly to \((2, 2)\). This shows the tail-to-tip method and confirms that \( \mathbf{u} + \mathbf{v} = \left[ \begin{array}{c} 2 \ 2 \end{array} \right] \).

Key Concepts

Graphical Representation of Vectors

When working with vectors, visualizing them graphically can greatly enhance understanding. Imagine the XY plane as a blank canvas where every vector can be drawn as an arrow. The origin, point (0,0), is the starting point for our arrows.

For vector \( \mathbf{u} \), represented by \( [3, 4] \), we start drawing an arrow from the origin to the point (3,4). This arrow illustrates the direction and magnitude of \( \mathbf{u} \). Similarly, vector \( \mathbf{v} \) begins at the origin and points towards (-1, -2). However, when performing vector addition, we don't place every vector at the origin.

Instead, we utilize the tail-to-tip method. After plotting \( \mathbf{u} \), move to the tip of \( \mathbf{u} \) at (3,4) and start \( \mathbf{v} \) from there. This stepwise connection of vectors provides clear visual evidence of the operation and helps conclude the resultant vector \( \mathbf{u} + \mathbf{v} \) with ease, which in this exercise is \( [2, 2] \). By using graphical techniques alongside arithmetic solutions, we reinforce understanding with every vector operation.

Tail-to-Tip Method

The tail-to-tip method is a straightforward and valuable way to add vectors visually. It emphasizes the continuous path from one vector to the next, by connecting them in order.

To apply this method, start with the first vector, drawing it from the origin to its specific coordinates. When the first vector ends, the second vector begins, with its tail at the tip of the first.

Consider \( \mathbf{u} + \mathbf{v} \). Draw \( \mathbf{u} \) from (0,0) to (3,4), then draw \( \mathbf{v} \) from (3,4) to (2,2). This continuous line from the origin to \( (2, 2) \) represents the resultant vector. This method not only aids in visualizing results but also provides a geometric understanding of vector operations.

Component-Wise Addition

Component-wise addition is a basic but essential skill in vector mathematics. It involves adding corresponding components of two vectors, thus simplifying vector addition.

For vectors \( \mathbf{u} = [3, 4] \) and \( \mathbf{v} = [-1, -2] \), the process involves simple arithmetic:

• First component: \( 3 + (-1) = 2 \)
• Second component: \( 4 + (-2) = 2 \)

These calculations produce the resultant vector \( [2, 2] \). By understanding the arithmetic in component-wise addition, you gain insight into vector properties and operations. This breakdown not only simplifies calculations but also bridges the gap between numerical and graphical vector representations.