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

Step-by-Step Solution

Answer

The vector \( a\mathbf{x} = \begin{bmatrix} -1 \\ 3 \end{bmatrix} \) is one-third the length of \( \mathbf{x} \) and points in the opposite direction.

Step 1: Compute the Scalar Multiplication

Given the vector \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) and the scalar \( a = -\frac{1}{3} \), compute \( a\mathbf{x} \). Multiply each component of \( \mathbf{x} \) by \( a \): \[ a\mathbf{x} = -\frac{1}{3} \begin{bmatrix} 3 \ -9 \end{bmatrix} = \begin{bmatrix} -\frac{1}{3} \times 3 \ -\frac{1}{3} \times (-9) \end{bmatrix} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \] Thus, \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \).

Step 2: Represent Vectors on the Plane

Represent the original vector \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) and the new vector \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \) on the Cartesian plane. The vector \( \mathbf{x} \) points from the origin to the point (3, -9), and \( a\mathbf{x} \) points to (-1, 3).

Step 3: Graphical Explanation of Scalar Multiplication

To graphically understand \( a\mathbf{x} \), start at the origin (0,0) and draw the vector \( \mathbf{x} \) towards (3, -9). The effect of multiplying by \( a = -\frac{1}{3} \) contracts \( \mathbf{x} \) by one-third of its length and reverses its direction, resulting in the vector pointing towards (-1, 3). This demonstrates the geometric interpretation of scalar multiplication: it alters the vector's scale and direction.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in linear algebra. It involves taking a vector and multiplying it by a scalar, which is simply a real number. This operation modifies the vector by scaling its magnitude and possibly changing its direction. For instance, when you have a vector like \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) and a scalar \( a = -\frac{1}{3} \), you perform scalar multiplication by multiplying each component of the vector by the scalar.

• This results in a new vector \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \).
• The first component \( -1 \) is the product of \( 3 \) and \(- \frac{1}{3}\), whereas the second component \( 3 \) results from multiplying \(-9\) by \(- \frac{1}{3}\).

This illustrates how scalar multiplication can adjust both the size and direction of a vector, providing a deeper understanding of vectors in the context of linear algebra.

Vector Representation

Vector representation on a Cartesian plane is a powerful way to visualize vectors. Each vector in a 2D space is represented as an arrow that starts at the origin and ends at the coordinates specified by the vector's components.

• For instance, the vector \( \mathbf{x} = \begin{bmatrix} 3 \ -9 \end{bmatrix} \) points from the origin to the point (3, -9).
• After scalar multiplication, the vector \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \) points to (-1, 3).

The length and direction of the arrow signify the vector's magnitude and direction, respectively. This visual tool helps in understanding how scalar multiplication affects vectors, as seen with how the vector's direction is reversed and its length is scaled when multiplied by a negative scalar.

Cartesian Plane

The Cartesian plane is an essential tool in mathematics for plotting data and understanding vector behavior. It consists of two perpendicular axes, the x-axis and the y-axis, which form a flat plane. Each point on this plane is represented by an ordered pair of numbers (x, y).

• In vector representation, the plane allows you to plot the vector and its transformations visually.
• The vector \( \mathbf{x} \) pointing to (3, -9) illustrates its original placement on the plane.
• After applying the scalar \( a = -\frac{1}{3} \), this vector transforms to \( a\mathbf{x} = \begin{bmatrix} -1 \ 3 \end{bmatrix} \), thus pointing to (-1, 3).

The Cartesian plane is fundamental in showing how operations like scalar multiplication modify a vector's position and orientation. This method provides a clear and intuitive understanding of vector transformations in linear algebra.