Problem 23 - Calculus for Biology and Medicine

Question

In Problems \(23-28\), 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 \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} -2 \\ 1 \end{array}\right] \text { and } a=2 $$

Step-by-Step Solution

Answer

The scaled vector is \( a\mathbf{x} = \begin{bmatrix} -4 \\ 2 \end{bmatrix} \), which is twice as long as \( \mathbf{x} \).

Step 1: Understand the Problem

We are given a vector \( \mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \) and a scalar \( a = 2 \). Our task is to compute the vector \( a\mathbf{x} \), represent both \( \mathbf{x} \) and \( a\mathbf{x} \) on the plane, and provide a graphical explanation of the relationship between these vectors.

Step 2: Calculate the Scaled Vector

To find \( a\mathbf{x} \), multiply each component of \( \mathbf{x} \) by the scalar \( a \). Therefore, \[ a\mathbf{x} = 2 \cdot \begin{bmatrix} -2 \ 1 \end{bmatrix} = \begin{bmatrix} 2 \cdot (-2) \ 2 \cdot 1 \end{bmatrix} = \begin{bmatrix} -4 \ 2 \end{bmatrix}. \]

Step 3: Graphical Representation and Explanation

To graphically represent \( \mathbf{x} \) and \( a\mathbf{x} \), plot both vectors starting from the origin point (0,0) on a coordinate plane. The vector \( \mathbf{x} \) points to (-2,1), while the vector \( a\mathbf{x} \) points to (-4,2). Graphically, scaling a vector by a scalar results in the same direction (unless the scalar is negative, in which case the direction is reversed) but with a different length. Here, \( a = 2 \) doubles the length of the vector \( \mathbf{x} \) while maintaining its direction.

Key Concepts

Graphical Representation

When we talk about graphical representation in the context of vectors, we're essentially talking about plotting vectors on a grid to visually understand their properties like direction and magnitude. Imagine a grid similar to a chessboard but with much finer lines. Each square on this grid can be seen as representing a unit measure for both the x-axis and the y-axis.

• The vector \( \mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \) needs to be shown graphically. You see this as an arrow that starts at \( (0, 0) \) and points to \((-2, 1)\). This means you move two units to the left on the x-axis and one unit up on the y-axis.
• Now, for the vector \( a\mathbf{x} = \begin{bmatrix} -4 \ 2 \end{bmatrix} \), it's like you're stretching the original vector by a factor of two. So, you now move four units to the left and two units up.
• The entire process gives us a visual clue about how the vector is getting bigger (or smaller) when multiplied by a scalar.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot points, lines, and vectors to study their relationships. It's divided into four quadrants by the intersection of two perpendicular lines called axes.

• The horizontal line is the x-axis, and it represents left and right directions.
• The vertical line is the y-axis, which indicates up and down directions.
• In this exercise, you start plotting from the origin, which is the point \((0, 0)\). This is where the x-axis and y-axis meet.
• The coordinate plane allows us to track changes visually when we multiply vectors by scalars. The arrows representing vectors show their direction from the origin.

As you multiply a vector by a scalar, watch how the arrow representing the vector extends in length but keeps heading in the same overall direction. This unique feature of vector scaling is best appreciated on a coordinate plane.

Scalar Multiplication

Scalar multiplication is a fundamental concept when working with vectors, involving multiplying a vector by a single number (the scalar). This technique scales the vector, adjusting its size while maintaining its direction or flipping it if the scalar is negative.

• To understand this in action, take the vector \( \mathbf{x} = \begin{bmatrix} -2 \ 1 \end{bmatrix} \) and scalar \( a = 2 \).
• Multiply each component of \( \mathbf{x} \) by the scalar to get \( a\mathbf{x} \). This means doing \( 2 \times (-2) = -4 \) for the x-component and \( 2 \times 1 = 2 \) for the y-component.
• The result is a new vector \( a\mathbf{x} = \begin{bmatrix} -4 \ 2 \end{bmatrix} \). This new vector is twice as long as \( \mathbf{x} \) because the scalar is 2.
• In scalar multiplication, unless the scalar is negative, the direction stays the same. However, if it's negative, the direction flips, turning the vector around to point in the exact opposite direction.

This process underpins many practical applications, from physics to computer graphics, helping us model scenarios where changing magnitudes are a factor.