Problem 27 - 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 \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} -4 \\ 1 \end{array}\right] \text { and } a=1 / 4 $$

Step-by-Step Solution

Answer

The vector \( a \mathbf{x} \) is \( \begin{bmatrix} -1 \\ \frac{1}{4} \end{bmatrix} \). It represents \( \mathbf{x} \) scaled down by a factor of \( \frac{1}{4} \).

Step 1: Multiply the Vector by the Scalar

The given vector is \( \mathbf{x} = \begin{bmatrix} -4 \ 1 \end{bmatrix} \) and the scalar is \( a = \frac{1}{4} \). To compute \( a \mathbf{x} \), multiply each component of \( \mathbf{x} \) by \( a \). This gives: \[a \mathbf{x} = \frac{1}{4} \begin{bmatrix} -4 \ 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{4} \times -4 \ \frac{1}{4} \times 1 \end{bmatrix} = \begin{bmatrix} -1 \ \frac{1}{4} \end{bmatrix} \]

Step 2: Represent the Vectors on the Plane

To visualize the vectors, plot \( \mathbf{x} = \begin{bmatrix} -4 \ 1 \end{bmatrix} \) and \( a \mathbf{x} = \begin{bmatrix} -1 \ \frac{1}{4} \end{bmatrix} \) on a coordinate plane. The initial vector \( \mathbf{x} \) starts at the origin and ends at the point \( (-4, 1) \). The vector \( a \mathbf{x} \) also starts at the origin but ends at the point \( (-1, \frac{1}{4}) \).

Step 3: Graphically Explain the Scalar Multiplication

When you multiply a vector by a scalar \( \frac{1}{4} \), each component of the vector becomes a quarter of its original value. In a graphical sense, \( a \mathbf{x} \) is a scaled-down version of \( \mathbf{x} \), maintaining the same direction but with reduced magnitude. The vector \( a \mathbf{x} \) is parallel to \( \mathbf{x} \) and points in the same direction, as the scalar is positive.

Key Concepts

Vector Representation

Vectors are mathematical entities that have both magnitude and direction. They are often represented as arrows in the coordinate plane. In mathematical notation, vectors are typically expressed as column matrices, where each entry corresponds to a component of the vector along the respective axis. For example, the vector \( \mathbf{x} = \begin{bmatrix} -4 \ 1 \end{bmatrix} \) has a horizontal component of \(-4\) and a vertical component of \(1\).

• This means that starting from the origin, the vector points 4 units to the left and 1 unit up.
• Vectors can be used to represent physical quantities such as force, velocity, and displacement.

Understanding vector representation is fundamental in physics and engineering as it helps visualize and solve problems involving direction and magnitude.

Graphic Visualization

Graphic visualization of vectors is an essential part of understanding their properties and operations like addition, subtraction, and scalar multiplication. When visualized graphically, a vector is drawn as an arrow pointing from the origin to its endpoint, defined by its coordinates.

• The length of the arrow represents the magnitude of the vector.
• The angle the arrow makes with a reference axis represents the direction.
• In our problem, the vector \( \mathbf{x} = \begin{bmatrix} -4 \ 1 \end{bmatrix} \) is visualized from the origin to the point \((-4, 1)\) with an arrow.

This visual approach not only helps in understanding the vector itself but also aids in comprehending the effects of operations such as scaling, as they transform the vector in predictable ways.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates, usually in the form \((x, y)\). These points allow for precise placement and representation of vectors.

• The horizontal axis (often called the x-axis) represents horizontal movement.
• The vertical axis (y-axis) represents vertical movement.
• On this plane, vectors can be manipulated, and their transformations, such as rotations and scalings, are visually represented.

In our example, plotting the initial vector \( \mathbf{x} = \begin{bmatrix} -4 \ 1 \end{bmatrix} \) and the scalar-adjusted vector \( a \mathbf{x} = \begin{bmatrix} -1 \ \frac{1}{4} \end{bmatrix} \) allows us to visually track how the vector is scaled by a factor, in this case, \( \frac{1}{4} \). The vector endpoints indicate their positions on the coordinate plane, making the change due to scalar multiplication clear.

Scaling Vectors

Scaling a vector involves multiplying each of its components by a scalar value. This operation alters the magnitude of the vector without changing its direction if the scalar is positive.

• A scalar greater than 1 will stretch the vector, increasing its length.
• A scalar less than 1 will compress the vector, reducing its length.
• For negative scalars, the vector’s direction is reversed.

In the given exercise, the scalar is \( a = \frac{1}{4} \), which scales down the vector \( \mathbf{x} = \begin{bmatrix} -4 \ 1 \end{bmatrix} \) resulting in a new vector \( a \mathbf{x} = \begin{bmatrix} -1 \ \frac{1}{4} \end{bmatrix} \). This means the magnitude of the vector is reduced to a quarter of its original size, but the vector still points in the same direction, illustrating how scaling affects vector magnitude while maintaining its direction.