Problem 33 - Calculus for Biology and Medicine

Question

Let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$ Compute \(\mathbf{u}+\mathbf{v}+\mathbf{w}\) and illustrate the result graphically.

Step-by-Step Solution

Answer

The sum of \( \mathbf{u} + \mathbf{v} + \mathbf{w} \) is \( \begin{bmatrix} 3 \\ 0 \end{bmatrix} \).

Step 1: Add Vectors \( \mathbf{u} \) and \( \mathbf{v} \)

First, we need to add the vectors \( \mathbf{u} \) and \( \mathbf{v} \). The sum of two vectors \( \mathbf{a} = \begin{bmatrix} a_1 \ a_2 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} b_1 \ b_2 \end{bmatrix} \) is done component-wise:\[\mathbf{u} + \mathbf{v} = \begin{bmatrix} 3 \ 4 \end{bmatrix} + \begin{bmatrix} 1 \ -2 \end{bmatrix} = \begin{bmatrix} 3 + 1 \ 4 + (-2) \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix}\]

Step 2: Add the Result to Vector \( \mathbf{w} \)

Now, we add the result from Step 1 to the vector \( \mathbf{w} \):\[(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \begin{bmatrix} 4 \ 2 \end{bmatrix} + \begin{bmatrix} -1 \ -2 \end{bmatrix} = \begin{bmatrix} 4 + (-1) \ 2 + (-2) \end{bmatrix} = \begin{bmatrix} 3 \ 0 \end{bmatrix}\]

Step 3: Graphically Illustrate the Result

The vector \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \begin{bmatrix} 3 \ 0 \end{bmatrix} \) can be represented graphically as an arrow starting at the origin \((0, 0)\) and pointing to the point \((3, 0)\) on a 2D Cartesian plane.

Key Concepts

Understanding 2D Vectors

A 2D vector is like a line with direction and length confined to a flat plane. This plane, known as the Cartesian plane, has two lines that meet at right angles called axes. Each axis is numbered and shows how far something is along that direction. In terms of vectors, they are represented using coordinates, often noted as \( \begin{bmatrix} x \ y \end{bmatrix} \), where \( x \) is how far the vector moves along the horizontal (X) axis and \( y \) is how far it moves along the vertical (Y) axis.

Vectors are notorious for showing up in different fields like physics, engineering, and even computer graphics because they can efficiently represent magnitude and direction. In everyday terms, think of a 2D vector as an arrow. The arrow itself points in a specific direction and has a length, which represents the vector's magnitude. So, if you imagine our exercise, the vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) are these arrows on the plane, each pointing a certain way from the origin, (0, 0).

Graphical Representation of Vectors

Graphically, vectors are depicted on a Cartesian coordinate system as arrows. The start of the arrow is called the tail and the endpoint is the tip or head. The length of the arrow indicates the magnitude, while the direction in which the arrow points represents its direction. The likes of vectors \( \mathbf{u} \) with coordinates \( [3, 4] \), \( \mathbf{v} \) as \( [1, -2] \), and \( \mathbf{w} \) which is \( [-1, -2] \) all find their placement through the Cartesian plane.

To capture a vector like \( \mathbf{u} \), imagine starting from the origin (0, 0), moving 3 units along the X-axis, and 4 units upward along the Y-axis. Each vector is drawn as a straight line from the origin to the point it leads to, like \( (3, 4) \) for \( \mathbf{u} \). To illustrate the resultant vector \( \mathbf{u} + \mathbf{v} + \mathbf{w} = [3, 0] \), simply draw an arrow from the origin directly to the point (3, 0) on the plane.

• Origin: The starting point (0,0).
• X-axis: Horizontal line showing distance.
• Y-axis: Vertical line showing distance.
• Arrow: Depiction of vector's magnitude and direction.

Component-wise Addition of Vectors

Adding vectors in 2D is simple when you understand component-wise addition. Each vector has components, and adding them means summing the individual components of each vector. For example, if \( \mathbf{a} = \begin{bmatrix} a_1 \ a_2 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} b_1 \ b_2 \end{bmatrix} \), then \( \mathbf{a} + \mathbf{b} = \begin{bmatrix} a_1 + b_1 \ a_2 + b_2 \end{bmatrix} \). This operation essentially adds together the X-axis movements and the Y-axis movements separately.

In our exercise, to find \( \mathbf{u} + \mathbf{v} + \mathbf{w} \), you would first add \( \mathbf{u} \) and \( \mathbf{v} \) as \( \begin{bmatrix} 3 + 1 \ 4 - 2 \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix} \). Next, this result was added to \( \mathbf{w} \), yielding \( \begin{bmatrix} 4 - 1 \ 2 - 2 \end{bmatrix} = \begin{bmatrix} 3 \ 0 \end{bmatrix} \).

This method is particularly effective because you handle each axis independently, making it more intuitive. It mirrors real-life scenarios, like walking—where moving north-south (Y-axis) is treated separately from moving east-west (X-axis).

• Effective for 2D motion understanding
• Intuitive and insightful
• Works separately in X and Y directions