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

Step-by-Step Solution

Answer

The vector \( a\mathbf{x} = \left[ -1, \frac{1}{4} \right] \). It contracts towards the origin by \( \frac{1}{4} \).

Step 1: Identify the Vector and Scalar

Given the vector \( \mathbf{x} = \left[ \begin{array}{r} -4 \ 1 \end{array} \right] \) and the scalar \( a = \frac{1}{4} \). The task is to calculate the product \( a\mathbf{x} \).

Step 2: Compute the Scaled Vector

To compute \( a\mathbf{x} \), multiply each component of the vector \( \mathbf{x} \) by the scalar \( a \).Compute as follows:\[a\mathbf{x} = \frac{1}{4} \left[ \begin{array}{r} -4 \ 1 \end{array} \right] = \left[ \begin{array}{r} \frac{1}{4} \times (-4) \ \frac{1}{4} \times 1 \end{array} \right] = \left[ \begin{array}{r} -1 \ \frac{1}{4} \end{array} \right] \]

Step 3: Graphical Representation of Vectors

In the coordinate plane, represent the original vector \( \mathbf{x} \) by drawing an arrow from the origin to the point \((-4, 1)\). The scaled vector \( a\mathbf{x} \) is drawn from the origin to the point \((-1, \frac{1}{4})\), which is a contraction towards the origin by a factor of \( \frac{1}{4} \).

Step 4: Explanation of Computation and Graphical Observation

Multiplying a vector by a scalar affects its magnitude but not its direction. Here, scaling \( \mathbf{x} \) by \( \frac{1}{4} \) reduces its magnitude by a factor of \( \frac{1}{4} \), making the vector shorter and closer to the origin in the same direction.

Key Concepts

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar, which is simply a single number. This changes the size, or magnitude, of the vector but does not alter its direction unless the scalar is negative. For example, if you have a vector \( \mathbf{x} = \left[ \begin{array}{r} -4 \ 1 \end{array} \right] \) and a scalar \( a = \frac{1}{4} \), multiplying them gives you a new vector:

• Multiply each component separately: \( a\mathbf{x} = \left[ \begin{array}{r} \frac{1}{4} \times -4 \ \frac{1}{4} \times 1 \end{array} \right] \).
• Result: \( a\mathbf{x} = \left[ \begin{array}{r} -1 \ \frac{1}{4} \end{array} \right] \).

This operation scales the vector by the factor of \( \frac{1}{4} \), making it shorter but pointing in the same direction. If the scalar was a negative number, the direction would be flipped, pointing in the opposite direction.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point on this plane is given by a pair of numbers \((x, y)\) corresponding to the coordinates in these directions.
When we apply scalar multiplication and draw our vectors on this plane, it visualizes how scaling affects them. Using vector \( \mathbf{x} = [-4, 1] \), it appears as an arrow from the origin \((0,0)\) to \((-4,1)\).
After scaling by \( a = \frac{1}{4} \), our new vector \( a\mathbf{x} \) ends at \((-1, \frac{1}{4})\). This shows the same vector shrunk by the scalar factor. The arrow remained in the same direction, simply getting shorter. This visual demonstrates the impact of scalar multiplication, making it easier to understand through graphical representation.

Vector Representation

A vector is often represented as an arrow on the coordinate plane, which clearly shows its direction and magnitude. The process of vector representation helps to visualize how vectors behave under certain operations like scaling.
For the vector \( \mathbf{x} = [-4, 1] \), its representation starts from the origin \((0, 0)\) to the point \((-4, 1)\). After performing scalar multiplication with \( a = \frac{1}{4} \), the new vector \( a\mathbf{x} \) is positioned from the origin to \((-1, \frac{1}{4})\).
Imaging these vectors as arrows helps us see the contraction towards the origin caused by the scalar. The visual aspect underscores how the vector's length changes, but its trajectory remains constant, unless the scalar is negative. This kind of representation is crucial in many fields, like physics and engineering, where direction and magnitude play a significant role in analysis and application.