Problem 30 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The vector \( \mathbf{u} - \mathbf{v} \) is \( \begin{bmatrix} 4 \\ 6 \end{bmatrix} \).

Step 1: Identify the Vectors

We are given three vectors: \( \mathbf{u} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \), \( \mathbf{v} = \begin{bmatrix} -1 \ -2 \end{bmatrix} \), and \( \mathbf{w} = \begin{bmatrix} 1 \ -2 \end{bmatrix} \). Our task is to compute \( \mathbf{u} - \mathbf{v} \).

Step 2: Compute the Difference of the Vectors

To find \( \mathbf{u} - \mathbf{v} \), subtract corresponding components of vector \( \mathbf{v} \) from vector \( \mathbf{u} \):\[\mathbf{u} - \mathbf{v} = \begin{bmatrix} 3 \ 4 \end{bmatrix} - \begin{bmatrix} -1 \ -2 \end{bmatrix} = \begin{bmatrix} 3 - (-1) \ 4 - (-2) \end{bmatrix} = \begin{bmatrix} 4 \ 6 \end{bmatrix}\]

Step 3: Illustrate the Result Graphically

To graphically illustrate \( \mathbf{u} - \mathbf{v} \), plot the vector \( \begin{bmatrix} 4 \ 6 \end{bmatrix} \) starting from the origin on a grid. The vector ends at the point (4, 6), and its components represent its movement from the origin: 4 units in the x-direction and 6 units in the y-direction.

Key Concepts

Vector Operations

Vectors are fundamental in mathematics, especially in physics and engineering, because they represent both magnitude and direction. One of the core operations involving vectors is subtraction. Vector subtraction is similar to regular subtraction but specific to vector components. For instance, if you have two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), the subtraction \( \mathbf{u} - \mathbf{v} \) involves taking each component of \( \mathbf{v} \) and subtracting it from the corresponding component of \( \mathbf{u} \).
In our exercise, we are given \( \mathbf{u} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} -1 \ -2 \end{bmatrix} \). The subtraction \( \mathbf{u} - \mathbf{v} \) is calculated as follows:

• Subtract the x-components: \( 3 - (-1) = 4 \)
• Subtract the y-components: \( 4 - (-2) = 6 \)

This results in the vector \( \begin{bmatrix} 4 \ 6 \end{bmatrix} \), which represents the new position in space following the subtraction.

Graphical Representation

Graphical representation of vectors is a powerful way to understand vector operations visually. When subtracting vectors, like \( \mathbf{u} - \mathbf{v} \), the result can be plotted on the Cartesian coordinate system to better visualize the change.
To represent \( \begin{bmatrix} 4 \ 6 \end{bmatrix} \) graphically, start at the origin (0,0) since vector operations typically begin from this point. Move 4 units right along the x-axis, then 6 units upwards along the y-axis. Mark the endpoint at coordinates (4, 6).
The entire vector can be seen as an arrow starting from the origin and ending at this point, clearly showing the direction and the magnitude of the resultant vector. The horizontal movement signifies change along the x-axis, and the vertical represents change along the y-axis.
By graphing vector results, students can intuitively grasp how vector operations affect both direction and position. It is like painting a path of where the vector takes you.

Mathematics Education

Understanding vectors is a substantial part of mathematics education, serving as a bridge between algebra and geometry. It plays a crucial role in developing spatial reasoning and problem-solving skills.
Learners are often introduced to vectors through graphical means first, which allows them to see the practical applications and benefits of using vectors. By visualizing operations like vector subtraction, students can better comprehend concepts like force, velocity, and other vector quantities in physics.
Clear teaching methods are essential. These include breaking down the operations:

• Identifying vector components
• Performing algebraic operations on each component
• Visualizing the final vector on a coordinate plane

Incorporating multiple examples and providing hands-on practice through exercises can deepen understanding and solidify this knowledge base.
Vector mathematics is a versatile and valuable tool, equipping students with the ability to tackle a range of real-world problems.