Problem 23 - Calculus for Biology and Medicine

Question

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}-2 \\ 1\end{array}\right]\) and \(a=2\)

Step-by-Step Solution

Answer

Scalar multiplication results in \( a\mathbf{x} = \begin{bmatrix} -4 \\ 2 \end{bmatrix} \), doubling the vector's length.

Step 1: Understanding the Vector

The given vector \( \mathbf{x} \) is \( \begin{bmatrix} -2 \ 1 \end{bmatrix} \). As it is in the form \( \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \), it corresponds to a point in the plane at coordinates (-2, 1).

Step 2: Applying Scalar Multiplication

To find \( a\mathbf{x} \), multiply each component of the vector \( \mathbf{x} \) by the scalar \( a \). Here, we compute: \( a\mathbf{x} = 2 \times \begin{bmatrix} -2 \ 1 \end{bmatrix} = \begin{bmatrix} 2 \times (-2) \ 2 \times 1 \end{bmatrix} = \begin{bmatrix} -4 \ 2 \end{bmatrix} \).

Step 3: Graphical Representation

On a coordinate plane, plot the original vector \( \mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \) starting from the origin (0,0) to the point (-2, 1). This is represented as an arrow pointing to (-2, 1). Next, plot the scaled vector \( a\mathbf{x} = \begin{bmatrix} -4 \ 2 \end{bmatrix} \) from the origin to the point (-4, 2). This new vector is parallel to \( \mathbf{x} \) and exactly twice as long, as expected due to the scalar multiplication.

Key Concepts

Vector Representation

Vectors are a fundamental concept in mathematics and physics, used to represent quantities that have both magnitude and direction. In this exercise, we examine the vector \( \mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \), which signifies a direction and magnitude on a two-dimensional plane.
To understand what a vector signifies, consider each component as a step in a specific direction:

• The first component, \(-2\), moves left on the horizontal axis.
• The second component, \(1\), moves upward on the vertical axis.

Thus, the vector \(\mathbf{x}\) effectively points from the origin \((0,0)\) to the coordinated point \((-2,1)\), with the line representing the vector.
Vectors provide a way to visually and mathematically model various phenomena, such as velocity, force, or any directional data. By understanding their components, we can determine both the direction and the length of their representation, critical in performing operations like scalar multiplication.

Coordinate Plane

The coordinate plane is a two-dimensional surface where vectors are visualized and other graphical operations are performed.
It consists of a horizontal x-axis and a vertical y-axis that intersect at a point called the origin, denoted as \((0,0)\). Vectors like \(\mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \) are plotted by moving from the origin according to their components.

• The coordinate \((-2, 1)\) indicates starting from the origin and moving 2 units to the left and 1 unit up.
• This results in the graphical depiction of the vector \(\mathbf{x}\) within this plane.

Understanding the coordinate plane is essential for visualizing vectors because it provides the framework to view and measure their properties such as direction, magnitude, and relative positioning. It serves as a backdrop for many mathematical concepts, enabling the visualization of abstract ideas like the effect of scalar multiplication on vectors.

Graphical Explanation

Graphical representations allow us to easily understand operations on vectors, such as scalar multiplication. When you multiply a vector by a scalar, you are essentially scaling its length while maintaining its direction.
In this exercise, we performed scalar multiplication of vector \(\mathbf{x}\) with \(a = 2\), resulting in \(a\mathbf{x} = \begin{bmatrix} -4 \ 2 \end{bmatrix} \).

• Initially, plot \(\mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \) on the coordinate plane to connect the origin and the point \((-2, 1)\).
• The scaled vector \(a\mathbf{x} = \begin{bmatrix} -4 \ 2 \end{bmatrix}\) should also begin from the origin, extending to the point \((-4, 2)\). This results in a new arrow twice the length of the original, but both lying on the same path.

Graphically, this manipulation shows up as a stretching of the initial vector. The direction remains unchanged, demonstrating that scalar multiplication compresses or stretches the vector along its line of action without altering its direction, essential for applications in physics and engineering.