Problem 25 - 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 \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}0 \\ -4\end{array}\right]\) and \(a=0.5\)

Step-by-Step Solution

Answer

The vector \(a\mathbf{x}\) is \(\begin{bmatrix} 0 \\ -2 \end{bmatrix}\).

Step 1: Understanding the Multiplication

To compute the vector \(a\mathbf{x}\), we need to multiply each component of the vector \(\mathbf{x}\) by the scalar \(a\). In this case, \(\mathbf{x} = \begin{bmatrix} 0 \ -4 \end{bmatrix}\) and \(a = 0.5\).

Step 2: Perform the Scalar Multiplication

Multiply each component of \(\mathbf{x}\) by \(a\):For the first component: \(0.5 \times 0 = 0\).For the second component: \(0.5 \times (-4) = -2\).Thus, \(a\mathbf{x} = \begin{bmatrix} 0 \ -2 \end{bmatrix}\).

Step 3: Graphical Representation

In the graph, 1. Represent the original vector \(\mathbf{x} = \begin{bmatrix} 0 \ -4 \end{bmatrix}\) by drawing an arrow from the origin (0,0) to the point (0, -4).2. Represent the new vector \(a\mathbf{x} = \begin{bmatrix} 0 \ -2 \end{bmatrix}\) from the origin to the point (0, -2).Since \(a = 0.5\), the new vector is 0.5 times the length of the original \(\mathbf{x}\) and points in the same direction.

Key Concepts

Scalar Multiplication

When working with vectors and scalars, scalar multiplication is a fundamental operation. Suppose you have a vector \( \mathbf{x} \) and a scalar \( a \). Scalar multiplication involves multiplying each component of the vector by the scalar. This results in a new vector pointing in the same direction but either longer or shorter, depending on the value of \( a \). For example, if \( a = 0.5 \), each component of \( \mathbf{x} \) is halved, creating a vector that is half the length of the original.

• If \( a = 1 \), the vector's magnitude remains unchanged.
• When \( a > 1 \), the vector's length increases.
• If \( a < 1 \), it becomes shorter.
• A negative \( a \) reverses the vector's direction.

Remember, each component of the vector is multiplied by the same scalar, which preserves the direction of the vector.

Vector Representation

Vectors can be represented in the plane using coordinates. In two dimensions, a vector \( \mathbf{x} \) like \( \begin{bmatrix} 0 \ -4 \end{bmatrix} \) represents a movement from the origin to the point \((0, -4)\) on a graph. This helps visualize vectors in a straightforward manner by portraying them as arrows. The tail of the arrow starts at the origin, and the head points towards the vector's terminal coordinates. The components of the vector define its direction and magnitude.

• The first component determines horizontal movement. Here it's \( 0 \), so there's no movement along the x-axis.
• The second component indicates vertical movement. In this example, it is \( -4 \), showing a downward direction along the y-axis.

Graphically representing vectors makes it easier to understand operations like scalar multiplication, whereby the entire arrow scales in size but keeps its direction unless the scalar is negative.

Graphical Vector Analysis

To understand graphical vector analysis, imagine drawing vectors in a coordinate system. For the vector \( \mathbf{x} = \begin{bmatrix} 0 \ -4 \end{bmatrix} \), start by drawing an arrow from the origin (0,0) to the point (0,-4). This arrow visually shows the vector's direction and magnitude.Next, when performing scalar multiplication with \( a \), a new vector is created. For instance, using \( a = 0.5 \), sketch an arrow from the origin to the point (0,-2).

• The direction stays the same as the original vector but its length is scaled by \( a \), which in this case is half the original length.
• This scaling effect due to \( a \) is what's visually evident on the graph.

The graphical representation emphasizes the scalar's impact by comparing the lengths and directions of both vectors, vividly showcasing how scalar multiplication transforms vectors in space. The result is an intuitive grasp of how vectors behave when scaled.