Problem 20 - Calculus for Biology and Medicine

Question

Find \(\mathbf{x}+\mathbf{y}\) for each pair of vectors \(\mathbf{x}\) and \(\mathbf{v}\). Represent \(\mathrm{x}, \mathbf{y}\), and \(\mathrm{x}+\mathrm{y}\) in the plane, and explain graphically how you add \(\mathbf{x}\) and \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}-1 \\ -1\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{l}1 \\ 2\end{array}\right]\)

Step-by-Step Solution

Answer

The sum of the vectors \( \mathbf{x} + \mathbf{y} \) is \( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \). Draw \( \mathbf{x} \) and \( \mathbf{y} \) and use the tip-to-tail method to visualize the vector addition.

Step 1: Understand the Vectors

Begin by identifying the given vectors. Here, we have two vectors: \( \mathbf{x} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \). Each vector has two components, corresponding to their x and y coordinates.

Step 2: Add the Vectors Algebraically

To find \( \mathbf{x} + \mathbf{y} \), add the corresponding components of the vectors. So, the x-components are \(-1 + 1 = 0\), and the y-components are \(-1 + 2 = 1\). Therefore, \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \).

Step 3: Graphical Representation of Vectors

Draw the initial point of \( \mathbf{x} \) at the origin (0,0) and point towards (-1,-1). Then plot \( \mathbf{y} \) starting at the origin as well, pointing towards (1,2).

Step 4: Visualizing the Vector Addition

Place the tail of \( \mathbf{y} \) at the head of \( \mathbf{x} \), which moves vector \( \mathbf{y} \) to be positioned with origin at (-1,-1). The resultant vector \( \mathbf{x} + \mathbf{y} \) is drawn from the tail of \( \mathbf{x} \) at the origin to the new head of \( \mathbf{y} \), ending at (0,1). This graphical method is known as the "tip-to-tail" method.

Key Concepts

Algebraic Addition of Vectors

Adding vectors algebraically involves straightforward addition of the corresponding components from each vector, like adding integers. Vectors are often represented in component form, \( \mathbf{x} = \begin{bmatrix} x_1 \ y_1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} x_2 \ y_2 \end{bmatrix} \), making it simple to compute their sum.

• Step 1: Identify the components of each vector.
• Step 2: Add the x-components of the vectors (i.e., \(x_1 + x_2\)).
• Step 3: Add the y-components of the vectors (i.e., \(y_1 + y_2\)).

For example, if \( \mathbf{x} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \), the sum \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \). This simple addition results in a new vector that is the algebraic sum of the two original vectors.

Graphical Representation of Vectors

Visualizing vectors on a coordinate plane offers a clear picture of how vectors can be added. When you draw vectors, you represent their magnitude and direction clearly. For \( \mathbf{x} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \), one would draw an arrow starting at the origin (0,0) and ending at the point (-1,-1). Similarly, for \( \mathbf{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \), the arrow starts at (0,0) and points to (1,2).

• Consistent Scaling: Ensure each unit on the axes represents the same length for accuracy.
• Arrowheads: Show the direction of the vector.

Graphically observing these vectors helps solidify the understanding of their directions and relative positions.

Tip-to-Tail Method

The "tip-to-tail" method is a fundamental approach for vector addition. It involves placing the tail of one vector at the tip (head) of another. This visual technique aids in understanding how vectors combine to form a resultant vector.

• Step 1: Begin by plotting \( \mathbf{x} \) from the origin.
• Step 2: Move \( \mathbf{y} \) such that its tail begins at the head of \( \mathbf{x} \).
• Step 3: Draw the resultant vector from the origin to the head of \( \mathbf{y} \).

This graphical illustration highlights that the new vector's path connects the starting point to the terminal point of the positioned \( \mathbf{y} \). In our example, the vector \( \mathbf{x} + \mathbf{y} \) is visualized from the start at (0,0) to end at (0,1). The tip-to-tail method not only shows the resultant vector's magnitude and direction but also reinforces understanding through visualization.