Problem 16 - Calculus for Biology and Medicine

Question

Find the dot product of \(\mathbf{x}=[-1,2]^{\prime}\) and \(\mathbf{y}=[-3,-4]^{\prime}\)

Step-by-Step Solution

Answer

The dot product of the vectors is -5.

Step 1: Understanding the Dot Product

The dot product of two vectors, \( \mathbf{x} = [x_1, x_2] \) and \( \mathbf{y} = [y_1, y_2] \), is calculated as \( x_1y_1 + x_2y_2 \). It's essentially a sum of the products of their corresponding elements.

Step 2: Identify the Components of Vectors

First, let's identify the components of the given vectors. The vector \( \mathbf{x} = [-1, 2]' \) has components \( x_1 = -1 \) and \( x_2 = 2 \). The vector \( \mathbf{y} = [-3, -4]' \) has components \( y_1 = -3 \) and \( y_2 = -4 \).

Step 3: Calculate Each Product

Now calculate the product of each corresponding components: \( x_1y_1 = (-1)(-3) = 3 \) and \( x_2y_2 = (2)(-4) = -8 \).

Step 4: Sum the Products

Add the results of the products from the previous step: \( 3 + (-8) = 3 - 8 = -5 \).

Step 5: Conclude the Dot Product

The dot product of the vectors \( \mathbf{x} \) and \( \mathbf{y} \) is thus \(-5\).

Key Concepts

Vectors

Imagine vectors as arrows in space, where each vector has both a direction and a magnitude. These can be in two dimensions like the example we have here, or even three or more dimensions in more complex cases. In simple terms, vectors are used to represent quantities that have a direction and a size, like wind speed or forces in physics.
In our scenario, the vectors

• \( \mathbf{x} = [-1, 2]' \) and
• \( \mathbf{y} = [-3, -4]' \)

are two-dimensional vectors. Each component of the vector describes its position along the respective axis in the coordinate system. These vectors can be visualized as points in a two-dimensional grid or plane, where the first number specifies the horizontal component, and the second gives the vertical component.
When we talk about vectors in mathematics and physics, we're often interested in operations like addition, subtraction, and multiplication, including the dot product which uses the elements of vectors to result in a scalar value.

Vector Components

When dissecting a vector, it's essential to understand its components, which are fundamental parts of the vector that provide its structure and orientation. These components simplify calculations and enable meaningful manipulations.
Consider the vector \( \mathbf{x} = [-1, 2]' \). Here,

• \( x_1 = -1 \) is the component along the x-axis, informing us how far left the vector points.
• \( x_2 = 2 \) tells us its upward direction along the y-axis.

Similarly, for \( \mathbf{y} = [-3, -4]' \):

• \( y_1 = -3 \) points to the left, and
• \( y_2 = -4 \) downward.

These components are crucial as they determine the vector's influence in its respective axes and are essential in operations like the dot product, where the corresponding components of vectors are multiplied and summed up.

Calculus

While the featured exercise focuses on a particular type of vector operation, calculus is a broader field that often employs vectors to solve complex problems. Calculus, in essence, examines how things change, using derivatives and integrals to understand rates and cumulative change.
Vectors, pivotal in calculus, help describe motion and forces in this context. For instance, the gradient vector is utilized in calculus to determine the direction and rate of fastest increase of a function in multivariable calculus. This, in turn, intertwines with the dot product, providing insights into relationships between varying quantities in space.
It's fascinating how dot products can arise naturally when optimally aligning vectors along with gradients, highlighting another powerful application of vectors in calculus. Even though this exercise looks at a straightforward dot product, these basic operations underpin critical concepts in higher-level calculus, illustrating broader applications of mathematical analysis.