Problem 26 - 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 \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}3 \\ -9\end{array}\right]\) and \(a=-1 / 3\)

Step-by-Step Solution

Answer

The vector \(a\mathbf{x}\) is \([-1, 3]\), reflecting \(\mathbf{x}\) and scaling it by \(-\frac{1}{3}\).

Step 1: Understand the exercise

The exercise asks us to compute the scalar multiplication of a vector \(\mathbf{x}\) by a scalar \(a\). The vector is \(\mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix}\) and the scalar is \(a = -\frac{1}{3}\). We need to find \(a\mathbf{x}\) and represent both \(\mathbf{x}\) and \(a\mathbf{x}\) in the coordinate plane.

Step 2: Scalar Multiplication

To compute \(a\mathbf{x}\), multiply each component of \(\mathbf{x}\) by \(a\). Thus, \[ a\mathbf{x} = -\frac{1}{3} \begin{bmatrix} 3 \ -9 \end{bmatrix} = \begin{bmatrix} -\frac{1}{3} \times 3 \ -\frac{1}{3} \times -9 \end{bmatrix} = \begin{bmatrix} -1 \ 3 \end{bmatrix}. \]

Step 3: Representation in the Plane

Now, we graphically represent \(\mathbf{x}\) and \(a\mathbf{x}\) on the coordinate plane. The vector \(\mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix}\) originates at the origin \((0,0)\) and points to \((3, -9)\). Similarly, the vector \(a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix}\) also starts at the origin and points to \((-1, 3)\).

Step 4: Graphical Explanation

Graphically, scalar multiplication by \(-\frac{1}{3}\) has the effect of scaling the vector \(\mathbf{x}\) down by a factor of \(\frac{1}{3}\) and also reverses its direction. This is visible in the points \(\begin{bmatrix} 3 \ -9 \end{bmatrix}\) being transformed to \(\begin{bmatrix} -1 \ 3 \end{bmatrix}\), effectively mirroring it across the origin.

Key Concepts

Scalar Multiplication

Scalar multiplication in vector calculus involves multiplying a vector by a scalar, which is simply a real number. This operation affects each component of the vector individually. Given a vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and a scalar \( a \), the product \( a\mathbf{x} \) is another vector, obtained by multiplying each component of \( \mathbf{x} \) by \( a \):

• \( a \times x_1 \) for the first component
• \( a \times x_2 \) for the second component

In our problem, we have \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) and \( a = -\frac{1}{3} \). The operation gives us \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \). It's important to notice how this scale down the vector length and can also reverse its direction if the scalar is negative.

Graphical Representation

Visualizing vectors on a graph can significantly help with understanding their properties and transformations. In vector calculus, each vector can be represented as an arrow pointing from the origin to a point corresponding to its components on a coordinate plane. For instance, vector \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) is a point at \( (3, -9) \), translating to an arrow from \( (0, 0) \) to \( (3, -9) \). Similarly, our transformed vector \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \) leads to an arrow reaching from the origin to \( (-1, 3) \).The change in vector direction in the graph can be attributed to the negative scalar, which flips the direction, and the length alteration draws from the absolute value of the scalar.

Coordinate Plane

The coordinate plane is a two-dimensional plane used extensively to plot vectors and perform mathematical operations like scalar multiplication. This plane comprises an x-axis (horizontal line) and a y-axis (vertical line). Each point on this plane denotes a vector with an x-component and a y-component. In our exercise, the original vector \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) is plotted at \( (3, -9) \). After scalar multiplication, the new vector \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \) appears at point \( (-1, 3) \). This simple visual manifestation can help illustrate operations like reflections and scaling, which occur during transformations. Such representation is a practical approach to understand and visualize vector transformations in mathematics.

Vector Transformation

Vector transformation encompasses any operation that changes a vector’s length, position, or direction. Scalar multiplication is a fundamental type of vector transformation widely used in calculus, physics, and engineering.

• A positive scalar stretches or compresses the vector.
• A negative scalar flips it to point in the opposite direction.

In this problem, applying scalar \( a = -\frac{1}{3} \) means that our vector \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) undergoes a transformation to become \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \). It vividly demonstrates the versatility and power of vectors by showing how simple scalar adjustments change their geometrical representation. Grasping these transformations can greatly aid in understanding more complex systems and equations encountered in advanced studies.