Problem 24 - 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 \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} 3 \\ -1 \end{array}\right] \text { and } a=-1 $$

Step-by-Step Solution

Answer

The vector \( a \mathbf{x} = \begin{bmatrix} -3 \\ 1 \end{bmatrix} \) reflects the original vector across the origin.

Step 1: Understand Vector Multiplication by a Scalar

When a vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) is multiplied by a scalar \( a \), each component of the vector is multiplied by \( a \). Here, the scalar \( a = -1 \).

Step 2: Calculate \( a \mathbf{x} \)

Given \( a = -1 \) and \( \mathbf{x} = \begin{bmatrix} 3 \ -1 \end{bmatrix} \), calculate the product as follows:\[a \mathbf{x} = -1 \cdot \begin{bmatrix} 3 \ -1 \end{bmatrix} = \begin{bmatrix} -1 \times 3 \ -1 \times (-1) \end{bmatrix} = \begin{bmatrix} -3 \ 1 \end{bmatrix}. \]

Step 3: Represent \( \mathbf{x} \) and \( a \mathbf{x} \) on the Plane

Plot \( \mathbf{x} \) as the point (3, -1) and \( a \mathbf{x} \) as (-3, 1) on a 2D coordinate plane. These points represent vectors stemming from the origin (0, 0) in respective directions.

Step 4: Explain Graphically the Effect of \( a \) on \( \mathbf{x} \)

Graphically, multiplying \( \mathbf{x} \) by \( -1 \) reflects it across the origin. The vector \( \begin{bmatrix} 3 \ -1 \end{bmatrix} \) is reflected to \( \begin{bmatrix} -3 \ 1 \end{bmatrix} \), maintaining the same magnitude but pointing in the opposite direction.

Key Concepts

Scalar Multiplication

Scalar multiplication is a key operation in vector algebra. It involves taking a vector and multiplying each of its components by a scalar, which is essentially just a real number. This is one of the foundational concepts in vector mathematics. For example, consider the vector \( \mathbf{x} = \begin{bmatrix} 3 \ -1 \end{bmatrix} \) and a scalar \( a = -1 \). When we multiply \( \mathbf{x} \) by \( a \), we multiply each component of \( \mathbf{x} \) by \( a \):

• The x-component (3) becomes \(-1 \times 3 = -3\).
• The y-component (-1) becomes \(-1 \times (-1) = 1\).

This results in a new vector: \( \begin{bmatrix} -3 \ 1 \end{bmatrix} \). Scalar multiplication alters the vector's direction and magnitude depending on the scalar's value.

2D Coordinate Plane

The 2D coordinate plane is a mathematical construct that's commonly used to visualize vector operations. It's made up of two perpendicular lines (axes) that intersect at a point called the origin (0,0):

• The horizontal axis is known as the x-axis.
• The vertical axis is called the y-axis.

Vectors in this plane are represented as arrows starting from the origin. For example, plotting the vector \( \mathbf{x} = \begin{bmatrix} 3 \ -1 \end{bmatrix} \) means drawing an arrow from (0,0) to (3,-1). Similarly, plotting \( a\mathbf{x} = \begin{bmatrix} -3 \ 1 \end{bmatrix} \) requires drawing an arrow from (0,0) to (-3,1). This graphical representation helps in understanding the changes vectors undergo during operations like scalar multiplication.

Vector Reflection

Vector reflection involves flipping a vector across an axis or the origin to achieve a mirror-image effect. When a vector is multiplied by the scalar \(-1\), it undergoes reflection in a 2D plane:

• Each component of the vector is negated.
• The direction of the vector is reversed, but its magnitude remains unchanged.

In our example, \( \mathbf{x} = \begin{bmatrix} 3 \ -1 \end{bmatrix} \) becomes \( a \mathbf{x} = \begin{bmatrix} -3 \ 1 \end{bmatrix} \) upon multiplication with \(-1\). On a coordinate plane, this intuitively appears as a flip through the origin. Such operations illustrate how scalars can distinctly alter vector orientations.

Vector Representation

Vector representation is crucial for visually and mathematically analyzing vectors in various fields, such as physics and engineering. Vectors are typically represented as an arrow or directed line segment in space. This representation includes both:

• The direction of the arrow, indicating where the vector points.
• The length of the arrow, representing the vector's magnitude.

For instance, \( \mathbf{x} = \begin{bmatrix} 3 \ -1 \end{bmatrix} \) is depicted as a directed line from the origin to the point (3, -1), while \( a\mathbf{x} = \begin{bmatrix} -3 \ 1 \end{bmatrix} \) extends to (-3, 1). These visual aids facilitate better comprehension of vector operations and their results by linking mathematical calculations to spatial understanding.