Problem 24 - 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 \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}3 \\ -1\end{array}\right]\) and \(a=-1\)

Step-by-Step Solution

Answer

The vector \( a\mathbf{x} \) is \( \left[ \begin{array}{r} -3 \\ 1 \end{array} \right] \), which is a reflection of \( \mathbf{x} \) through the origin.

Step 1: Understand Vector and Scalar Multiplication

We have a vector \( \mathbf{x} = \left[ \begin{array}{r} 3 \ -1 \end{array} \right] \) and a scalar \( a = -1 \). We need to compute the product \( a\mathbf{x} \) by multiplying each component of the vector \( \mathbf{x} \) by the scalar \( a \).

Step 2: Multiply Each Component

To find \( a\mathbf{x} \), multiply each component of the vector \( \mathbf{x} \) by \( a \). Thus, the first component: \( 3 \times (-1) = -3 \), and the second component: \( -1 \times (-1) = 1 \). The resulting vector is \( \mathbf{ax} = \left[ \begin{array}{r} -3 \ 1 \end{array} \right] \).

Step 3: Graphical Representation

Plot the vector \( \mathbf{x} \) in a coordinate plane starting from the origin \((0,0)\) and extending to \((3, -1)\). Next, plot the vector \( a\mathbf{x} = \left[ \begin{array}{r} -3 \ 1 \end{array} \right] \) by starting from the origin and extending to \((-3, 1)\). The transformation from \( \mathbf{x} \) to \( a\mathbf{x} \) is essentially a reflection through the origin, since the scalar \( a = -1 \) reverses the direction of each component of \( \mathbf{x} \).

Step 4: Interpretation of the Graphical Transformation

This multiplication demonstrates that a vector multiplied by a scalar \( a = -1 \) results in the vector pointing in the opposite direction. On the graph, this effect is visible as each vertex of \( \mathbf{x} \) being reflected through the origin to become the corresponding vertex of \( a\mathbf{x} \).

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar. A vector has both a direction and magnitude, represented with components, such as in the vector \( \mathbf{x} = \left[ \begin{array}{r} 3 \ -1 \end{array} \right] \). A scalar is simply a single number, like \( a = -1 \). When you perform scalar multiplication, you apply the scalar to each component of the vector separately. This means we multiply each element in the vector by the scalar. For instance:

• First component: \( 3 \times (-1) = -3 \)
• Second component: \( -1 \times (-1) = 1 \)

This results in a new vector, \( \mathbf{ax} = \left[ \begin{array}{r} -3 \ 1 \end{array} \right] \). Scalar multiplication is important in vector math because it scales or reflects vectors, allowing us to change vector sizes or directions.

Graphical Representation of Vectors

Understanding how vectors look in a plane helps us grasp vector operations, like scalar multiplication, visually. We can think of vectors as arrows pointing from the origin in the direction defined by their components. For the vector \( \mathbf{x} = \left[ \begin{array}{r} 3 \ -1 \end{array} \right] \), we draw an arrow starting from the origin \( (0,0) \) and ending at the point \( (3, -1) \).

• The tail of the vector is always at the origin.
• The head of the vector points to its components.

When scalars multiply vectors, the arrow's length changes or even reverses direction. The vector \( \mathbf{ax} = \left[ \begin{array}{r} -3 \ 1 \end{array} \right] \) from our example starts at the origin and ends at the point \( (-3, 1) \). This graphical approach illustrates transformations, by showing direction and magnitude changes.

Vector Transformation

Scalar multiplication can transform vectors in several ways, depending on the scalar's value. For example, multiplying by \( a = -1 \) inverts the vector. Let's consider how this transformation occurs:

• Direction: The vector's direction is reversed with negative scalars. In our example, \( \mathbf{x} = \left[ \begin{array}{r} 3 \ -1 \end{array} \right] \), transformed to \( \mathbf{ax} = \left[ \begin{array}{r} -3 \ 1 \end{array} \right] \), flips direction.
• Magnitude: The length of the vector remains the same when multiplying by -1, which reflects the vector across the origin. However, other scalars scale vectors proportionally.

These transformations are essential to understand directional changes and scaling in physics and engineering. They allow us to adjust the vectors' magnitude or completely reverse their path, adapting to different theoretical or real-world scenarios.