Problem 18 - 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 \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{r} -1 \\ 0 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

Step-by-Step Solution

Answer

\( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \).

Step 1: Understanding the Vectors

We are given two vectors \( \mathbf{x} \) and \( \mathbf{y} \). These vectors are written in component form where \( \mathbf{x} = \begin{bmatrix} -1 \ 0 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} 3 \ 2 \end{bmatrix} \). Each vector has two components: \( \mathbf{x} \) has components \(-1\) and \(0\), while \( \mathbf{y} \) has components \(3\) and \(2\). This indicates they are two-dimensional vectors on a plane.

Step 2: Performing Vector Addition

To find \( \mathbf{x} + \mathbf{y} \), we add the corresponding components of \( \mathbf{x} \) and \( \mathbf{y} \). The x-component of \( \mathbf{x} + \mathbf{y} \) is the sum of the x-components of \( \mathbf{x} \) and \( \mathbf{y} \): \( -1 + 3 = 2 \). The y-component is the sum of the y-components: \( 0 + 2 = 2 \). Hence, \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 2 \ 2 \end{bmatrix} \).

Step 3: Graphical Representation

To graph \( \mathbf{x}, \mathbf{y}, \) and \( \mathbf{x} + \mathbf{y} \) on the coordinate plane, plot each vector starting from the origin. Vector \( \mathbf{x} \) extends 1 unit to the left on the x-axis (due to \(-1\)) and does not move on the y-axis (\(0\)). Vector \( \mathbf{y} \) extends 3 units to the right and 2 units up. The resultant vector \( \mathbf{x} + \mathbf{y} \) goes from the origin to the point (2, 2), obtainable by the 'tip-to-tail' method, starting from the tip of \( \mathbf{x} \) to the tip of \( \mathbf{y} \).

Step 4: Explanation of Addition on the Plane

Vector addition can be visualized using the 'tip-to-tail' method on a graph. Place the tail of vector \( \mathbf{y} \) at the tip of vector \( \mathbf{x} \). Draw the resultant vector \( \mathbf{x} + \mathbf{y} \) starting from the tail of the first vector \( \mathbf{x} \) to the tip of the second vector \( \mathbf{y} \). This method visually illustrates vector addition, confirming that \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 2 \ 2 \end{bmatrix} \).

Key Concepts

Graphical Representation of Vectors

Visualizing vectors in a coordinate plane is an intuitive way of understanding what they represent. - Imagine a coordinate grid, similar to what you see in math class or on graph paper. - Vectors are represented as arrows on this grid, indicating direction and magnitude.- The origin of the arrow is where the vector "starts" and the arrowhead is where the vector "points". To graphically represent vector \(\mathbf{x} = \begin{bmatrix} -1 \ 0 \end{bmatrix}\), plot a point on the grid that is one unit left on the x-axis and does not move on the y-axis. ### How Does This Look?- Start by marking the origin, typically at the coordinates (0,0).- Move left along the x-axis to point (-1,0) and draw an arrow.- This is the vector \(\mathbf{x}\). For vector \(\mathbf{y} = \begin{bmatrix} 3 \ 2 \end{bmatrix}\), move right 3 units and up 2 units from the origin. - Plot the arrow's origin at (0,0) and tip at the point (3,2). Both arrows represent the initial position, direction, and magnitude of the vectors.

Two-Dimensional Vectors

Two-dimensional vectors provide information on a plane using two components: along the x-axis and the y-axis. - The notation \(\begin{bmatrix} a \ b \end{bmatrix}\) signifies a vector where \(a\) is the horizontal component and \(b\) is the vertical component. - In our case, vector \(\mathbf{x}\) with components \(-1\) and \(0\) shows movement purely along the negative x-axis. - Vector \(\mathbf{y}\), having components \(3\) and \(2\), makes a right-and-upward movement. Such vectors are known as **two-dimensional vectors** because they exist within this x-y plane. They allow us to represent a wide range of real-world phenomena such as movement, force, velocity, and more. ### Why Use Two-Dimensional Vectors?- These vectors simplify calculations like addition, subtraction, and scalar multiplication.- Understanding them helps in visualizing physical phenomena and in solving mathematical problems related to directions and magnitudes on a plane.

Tip-to-Tail Method

The 'tip-to-tail' method is a popular way to add vectors graphically, ensuring you understand how the resultant vector is formed.### How Does It Work?- Begin by plotting the first vector \(\mathbf{x}\). Draw it starting from the origin. - Once finished, take the tail of the next vector, in this case \(\mathbf{y}\), and place it at the tip of \(\mathbf{x}\). - The resultant vector, \(\mathbf{x} + \mathbf{y}\), is the vector drawn from the tail of \(\mathbf{x}\) (origin) to the tip of \(\mathbf{y}\).### Why Use This Method?- It offers a clear visual understanding of vector addition. - It's a straightforward approach suitable for visualizing how vectors combine their magnitudes and directions.- This makes realizing that the combined effect of \(\mathbf{x} = \begin{bmatrix} -1 \ 0 \end{bmatrix}\) and \(\mathbf{y} = \begin{bmatrix} 3 \ 2 \end{bmatrix}\) leads to \(\mathbf{x} + \mathbf{y} = \begin{bmatrix} 2 \ 2 \end{bmatrix}\) much more intuitive.