Problem 22 - 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 \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}-3 \\ -1\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{l}-1 \\ -1\end{array}\right]\)

Step-by-Step Solution

Answer

\( \mathbf{x} + \mathbf{y} = \begin{bmatrix} -4 \\ -2 \end{bmatrix} \). Graphically, add vectors head-to-tail to find the resultant.

Step 1: Understand Vector Notation

Vector notation \( \mathbf{x} = \begin{bmatrix} -3 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \) indicate that \( \mathbf{x} \) and \( \mathbf{y} \) are composed of two components, with the first component representing the horizontal direction (x-axis) and the second representing the vertical direction (y-axis).

Step 2: Add Vectors Component-wise

To find \( \mathbf{x} + \mathbf{y} \), add each corresponding component of \( \mathbf{x} \) and \( \mathbf{y} \):\[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} -3 \ -1 \end{bmatrix} + \begin{bmatrix} -1 \ -1 \end{bmatrix} = \begin{bmatrix} (-3) + (-1) \ (-1) + (-1) \end{bmatrix} = \begin{bmatrix} -4 \ -2 \end{bmatrix} \]This result shows the vector sum \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} -4 \ -2 \end{bmatrix} \).

Step 3: Graphical Representation of Vectors

Draw \( \mathbf{x} \), \( \mathbf{y} \), and \( \mathbf{x} + \mathbf{y} \) on a coordinate plane. Start \( \mathbf{x} \) at the origin and move \(-3\) units to the left and \(-1\) unit down. Similarly, for \( \mathbf{y} \), add it tail-to-head starting from the head of \( \mathbf{x} \) moving \(-1\) unit to the left and \(-1\) unit down.The final position reached is \( \mathbf{x} + \mathbf{y}: \begin{bmatrix} -4 \ -2 \end{bmatrix} \), which aligns with starting at the origin, moving \(-4\) units left and \(-2\) units down from the origin.

Step 4: Summary of Vector Addition

Graphically, adding vectors \( \mathbf{x} \) and \( \mathbf{y} \) involves placing them head-to-tail. The resultant vector \( \mathbf{x} + \mathbf{y} \) is the vector pointing from the origin to the endpoint of the second vector (\( \mathbf{y} \)), showing a new vector that represents the combined effect of \( \mathbf{x} \) and \( \mathbf{y} \).

Key Concepts

Component-wise Addition

Vector addition is a crucial concept in mathematics and physics. It involves combining two or more vectors to get a resultant vector. To perform vector addition, we use the method of component-wise addition. Simply put, this means adding the corresponding components of the vectors separately.

For example, given vectors \( \mathbf{x} = \begin{bmatrix} -3 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \), each vector has two components. The first component refers to the horizontal or x-direction, while the second indicates the vertical or y-direction.

To find the sum \( \mathbf{x} + \mathbf{y} \), add their respective x and y components:

• The x-components: \(-3 + (-1) = -4\)
• The y-components: \(-1 + (-1) = -2\)

Therefore, the resultant vector is \( \begin{bmatrix} -4 \ -2 \end{bmatrix} \). Each element in the resultant vector represents the combined effect of the respective components from the original vectors.

Graphical Representation

Graphical representation of vectors can greatly aid in understanding vector addition. When visualizing vectors, you can draw them on a coordinate plane. The process of adding vectors can be grasped intuitively by drawing them head-to-tail.

Let's take the vectors \( \mathbf{x} \) and \( \mathbf{y} \) from our example. To draw \( \mathbf{x} \), start from the origin, move 3 units to the left (since it’s negative) and 1 unit down. For \( \mathbf{y} \), begin from where \( \mathbf{x} \) ends, moving 1 unit more to the left and 1 unit further down.

• Start at the origin for \( \mathbf{x} \)
• Move left and down respecting the vector's components
• From the end of \( \mathbf{x} \), trace \( \mathbf{y} \)

The ending point after adding \( \mathbf{y} \) is where the resultant vector \( \begin{bmatrix} -4 \ -2 \end{bmatrix} \) ends. This point shows you visually how combining the two vectors affects the position on the plane.

Coordinate Plane

Using a coordinate plane simplifies understanding the position and direction of vectors. It's like having a grid map where you can plot points based on coordinates. With axes labeled x and y, you can easily visualize vectors like \( \mathbf{x} = \begin{bmatrix} -3 \ -1 \end{bmatrix} \).

The coordinate plane helps show how each vector moves from one point to another. The origin, usually at point \((0, 0)\), serves as a starting point to ensure the positions are clear and organized.

• The x-value indicates horizontal movement: left for negative, right for positive.
• The y-value signals vertical movement: down for negative, upward for positive.

When both vectors \( \mathbf{x} \) and \( \mathbf{y} \) are plotted on this plane, you can quickly see their impact. Each coordinate’s alteration is visible, making the concept of vector addition more tangible. Thus, the coordinate plane is a fundamental tool in vector operations, providing a visual aid to facilitate learning.