Problem 28 - Calculus for Biology and Medicine

Question

In Problems , compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{l} 0.5 \\ 0.25 \end{array}\right] \text { and } a=5 $$

Step-by-Step Solution

Answer

The scaled vector \( a \mathbf{x} \) is \( \begin{bmatrix} 2.5 \\ 1.25 \end{bmatrix} \), five times the length of \( \mathbf{x} \).

Step 1: Understand the Task

In this exercise, we are given a vector \( \mathbf{x} = \begin{bmatrix} 0.5 \ 0.25 \end{bmatrix} \) and a scalar \( a = 5 \). We need to compute the vector \( a \mathbf{x} \) by scaling \( \mathbf{x} \) with \( a \), and graphically represent both \( \mathbf{x} \) and \( a \mathbf{x} \) on the Cartesian plane.

Step 2: Compute the Scaled Vector

To find the vector \( a \mathbf{x} \), multiply each component of \( \mathbf{x} \) by \( a \). Perform the following calculations:\[ a \mathbf{x} = 5 \begin{bmatrix} 0.5 \ 0.25 \end{bmatrix} = \begin{bmatrix} 5 \times 0.5 \ 5 \times 0.25 \end{bmatrix} = \begin{bmatrix} 2.5 \ 1.25 \end{bmatrix} \]Thus, \( a \mathbf{x} = \begin{bmatrix} 2.5 \ 1.25 \end{bmatrix} \).

Step 3: Graphical Representation of Vectors

Plot the original vector \( \mathbf{x} = \begin{bmatrix} 0.5 \ 0.25 \end{bmatrix} \) on a 2D plane starting from the origin (0,0) to the point (0.5, 0.25). Next, plot the vector \( a \mathbf{x} = \begin{bmatrix} 2.5 \ 1.25 \end{bmatrix} \) starting from the origin to the point (2.5, 1.25). This vector is 5 times longer than \( \mathbf{x} \) and has the same direction.

Step 4: Explain the Vector Scaling Process

Graphically, when a vector is multiplied by a scalar \( a \), its length is scaled by the factor of \( a \) without changing its direction (only if \( a \) is positive). In this case, multiplying \( \mathbf{x} \) by 5 results in \( a \mathbf{x} \), which is stretched 5 times longer than \( \mathbf{x} \), pointing in the same direction as \( \mathbf{x} \).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in linear algebra and vector mathematics. It involves multiplying a vector by a scalar, which is simply a real number. The process of scalar multiplication scales the vector, meaning it changes its length while keeping its direction constant.

To perform scalar multiplication:

• Multiply each component of the vector by the scalar.
• The result is a new vector whose components are each scaled by the given scalar value.

For example, consider the vector \( \mathbf{x} = \begin{bmatrix} 0.5 \ 0.25 \end{bmatrix} \) and the scalar \( a = 5 \). The scaled vector \( a \mathbf{x} \) would be calculated as:\[ a \mathbf{x} = 5 \times \begin{bmatrix} 0.5 \ 0.25 \end{bmatrix} = \begin{bmatrix} 2.5 \ 1.25 \end{bmatrix} \]
Here, each component of \( \mathbf{x} \) is multiplied by 5, leading to the new vector \( \begin{bmatrix} 2.5 \ 1.25 \end{bmatrix} \), which is 5 times longer but in the same direction as \( \mathbf{x} \).

2D Vectors

2D vectors, or two-dimensional vectors, are basic elements of vector mathematics used to represent quantities that have both magnitude and direction in a two-dimensional space. Each vector consists of two components, typically represented as \( \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix} \), where \( x \) and \( y \) correspond to the horizontal and vertical components, respectively.

Understanding 2D vectors involves:

• Recognizing that they start from the origin or any other point in the plane, reaching out to another point defined by its components.
• Visualizing them as arrows in the Cartesian coordinate system, originating from \((0,0)\) to \((x,y)\).

For instance, the vector \( \mathbf{x} = \begin{bmatrix} 0.5 \ 0.25 \end{bmatrix} \) points from the origin to the point (0.5, 0.25) in the plane, defining both a direction and a magnitude or length.

Graphical Representation

Graphically representing vectors helps in understanding their nature and how scalar multiplication affects them. Vectors are often depicted as arrows on a Cartesian coordinate system.

Here's how to visually represent vectors and their scalar multiples:

• First, plot the original vector, which originates from the origin (0,0) to the terminal point defined by its components, such as \( (0.5, 0.25) \) for \( \mathbf{x} \).
• Next, plot its scalar multiple by using the same starting point, guiding the arrow to the new terminal point, which in our example is \( (2.5, 1.25) \) for \( a \mathbf{x} \).

Observing the plotted vectors, you'll notice that the arrow representing the scalar multiple is longer (or shorter if the scalar is between 0 and 1) yet maintains the same direction. If \( a < 0 \), the vector reverses its direction. This graphical representation effectively demonstrates that scalar multiplication scales the vector's magnitude while preserving (or reversing) its direction.