Problem 21 - 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 \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}1 \\ 0\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{r}-1 \\ 0\end{array}\right]\)

Step-by-Step Solution

Answer

\(\mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\), resulting in the zero vector.

Step 1: Identify the Vectors

We are given two vectors, \(\mathbf{x}\) and \(\mathbf{y}\), which are \(\mathbf{x} = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) and \(\mathbf{y} = \begin{bmatrix} -1 \ 0 \end{bmatrix}\), respectively. These vectors will be added together.

Step 2: Add the Vectors Algebraically

To find \(\mathbf{x} + \mathbf{y}\), we add the corresponding components of the vectors: \[\mathbf{x} + \mathbf{y} = \begin{bmatrix} 1 \ 0 \end{bmatrix} + \begin{bmatrix} -1 \ 0 \end{bmatrix} = \begin{bmatrix} 1 + (-1) \ 0 + 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}. \]

Step 3: Represent the Vectors on the Plane

Graphically, represent \(\mathbf{x}\) and \(\mathbf{y}\) as arrows starting from the origin on a coordinate plane. \(\mathbf{x}\) points rightwards along the x-axis with a magnitude of 1, and \(\mathbf{y}\) points leftwards along the x-axis with a magnitude of 1.

Step 4: Visualize Addition Graphically

When adding vectors graphically, place the tail of \(\mathbf{y}\) at the head of \(\mathbf{x}\). Since \(\mathbf{y}\) points in the opposite direction with the same magnitude, it exactly cancels \(\mathbf{x}\), ending back at the origin. The resultant vector, \(\mathbf{x} + \mathbf{y}\), is the zero vector \(\begin{bmatrix} 0 \ 0 \end{bmatrix}\).

Key Concepts

Graphical Representation of Vectors

Visualizing vectors on a coordinate plane can make understanding their properties much simpler. Vectors are often represented as arrows. This helps one see both direction and magnitude. Let's take the vectors from the exercise:

• Vector \(\mathbf{x}\): Starts at the origin \((0,0)\) and extends right on the x-axis, ending at the point \((1,0)\). This shows it has a magnitude of 1 in the positive x-direction.


• Vector \(\mathbf{y}\): Starts at the origin \((0,0)\) and extends left on the x-axis, ending at the point \((-1,0)\). Here, the magnitude is 1 but in the negative x-direction.

Representing vectors graphically allows us to observe the direction and strength (magnitude) of their effects, making it easier to see how they will interact when combined.

Resultant Vector

The resultant vector is what you get when you add two or more vectors together. It's like combining the forces or movements each vector represents into one overall effect.
To find the resultant graphically, align the tail of one vector to the head of the other. In our particular problem:

• Start with vector \(\mathbf{x}\), pointing right.


• Place the tail of vector \(\mathbf{y}\) at the head of \(\mathbf{x}\). Since \(\mathbf{y}\) points left, it brings us back to the origin.

Thus, the resultant is a vector that effectively doesn't change the point of origin, creating the zero vector. When adding vectors algebraically, as demonstrated, the components are summed directly, confirming the graphical intuition.

Zero Vector

The zero vector is a special type of vector represented by \(\begin{bmatrix} 0 \ 0 \end{bmatrix}\). It has zero magnitude and no specific direction. In the context of vector addition, it represents the concept of balance or cancellation.
Think of it this way: when vector \(\mathbf{x}\) moves right and vector \(\mathbf{y}\) moves left with equal force, their overall effect cancels out. That's why their sum is the zero vector.

• No magnitude: It does not move in any direction.


• No direction: Since it doesn't exert any influence, it lacks a direction.

In vector problems, encountering a zero vector often means either opposing vectors perfectly balance each other out, or there's simply no movement or change in that plane.