Problem 20 - Calculus for Biology and Medicine

Question

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{l} -1 \\ -1 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] $$

Step-by-Step Solution

Answer

\(\mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\).

Step 1: Identify the Vectors

Write down the given vectors \[ \mathbf{x} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \]and \[ \mathbf{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \].

Step 2: Perform Vector Addition

Add the corresponding components of vectors \(\mathbf{x}\) and \(\mathbf{y}\). For the first component: \(-1 + 1 = 0\), and for the second component: \(-1 + 2 = 1\).

Step 3: Write the Resultant Vector

The vector sum \(\mathbf{x} + \mathbf{y}\) is \[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \].

Step 4: Graphical Representation

Graph \(\mathbf{x}\) by plotting the point \((-1, -1)\) and \(\mathbf{y}\) by plotting the point \((1, 2)\). Begin at the origin, draw \(\mathbf{x}\) to \((-1, -1)\), then from that point, draw \(\mathbf{y}\) to \((0, 1)\). The endpoint represents the sum \((0, 1)\).

Key Concepts

Graphical Representation of Vectors

Vector addition can be visualized on a 2D plane, with each vector represented as an arrow starting from the origin. These arrows show both the direction and magnitude of the vector components.

For instance, with vectors \( \mathbf{x} = \begin{bmatrix} -1 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \):

• Start with vector \( \mathbf{x} \) at the origin, moving to point \((-1, -1)\).
• Then, place vector \( \mathbf{y} \) starting at the tip of vector \( \mathbf{x} \), leading to point \((0, 1)\).

This visual method is known as the "tip-to-tail" method, emphasizing the direction and magnitude of each vector component.
Graphically, vector addition is like drawing paths on a map with arrows. These arrows collectively represent both movement and resultant position in a plane.

Component-Wise Addition

Component-wise addition involves adding corresponding components of two or more vectors. This is vital for finding the resultant vector. Each vector's position in a plane contributes to the overall displacement.

Consider vectors \( \mathbf{x} \) and \( \mathbf{y} \):

• First component addition: \(-1 + 1 = 0\).
• Second component addition: \(-1 + 2 = 1\).

These sums form the new vector, often called the resultant vector. Understanding component-wise addition simplifies finding vector sums.
This approach allows one to break down complex vector issues into simpler arithmetic operations, making problem-solving clearer and more efficient.

Resultant Vector

The resultant vector is the outcome of adding two or more vectors together, providing a single vector that has the same effect as all the original vectors combined.

For vectors \( \mathbf{x} \) and \( \mathbf{y} \), the resultant vector \( \mathbf{x} + \mathbf{y} \) is found as:

• Calculating: \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \).
• Graphically, it's the endpoint reached after following both vectors from the origin.

The resultant vector helps in understanding cumulative effects and displacement caused by multiple vectors.
In contexts like physics and engineering, the resultant vector assists in determining overall direction and movement, essential for accurate and practical problem-solving.