Problem 96 - Algebra and Trigonometry

Question

Explaining the Concepts. If two vectors are expressed in terms of \(\mathbf{i}\) and \(\mathbf{j}\), explain how to find their difference.

Step-by-Step Solution

Answer

To find the difference of two vectors expressed in terms of \( \mathbf{i} \) and \( \mathbf{j} \), subtract the corresponding \( \mathbf{i} \) and \( \mathbf{j} \) components of the two vectors.

Step 1: Identification of Vectors

Identify the two vectors that are to be subtracted. Let's call these vectors A and B, and they can be represented as \(\mathbf{A} = x_1 \mathbf{i} + y_1 \mathbf{j}\) and \(\mathbf{B} = x_2 \mathbf{i} + y_2 \mathbf{j}\).

Step 2: Component-wise Subtraction

Subtract corresponding components of the two vectors. This is done by subtracting the \(\mathbf{i}\) component of vector B from the \(\mathbf{i}\) component of vector A and subtracting the \(\mathbf{j}\) component of vector B from the \(\mathbf{j}\) component of vector A. The resulting vector \(\mathbf{C} = (x_1 - x_2) \mathbf{i} + (y_1 - y_2) \mathbf{j}\) would be the difference between the two vectors A and B.

Key Concepts

Vector Components

Understanding vector components is fundamental to mastering vector operations such as vector subtraction. Vectors in a two-dimensional plane can be broken down into parts that run along the primary axes; in mathematical terminology, these parts are known as components. Each vector is expressed as a sum of its components along the x-axis and y-axis, typically written as \( \mathbf{i} \) and \( \mathbf{j} \), respectively. Imagine these components to be the projections of the vector onto the respective axes.

For instance, a vector \( \mathbf{A} \) represented as \( x_1 \mathbf{i} + y_1 \mathbf{j} \) has components \( x_1 \) and \( y_1 \) along the x-axis and y-axis, respectively. These numerical values indicate how far along each axis the vector extends. To find the difference between two vectors, one simply needs to subtract the corresponding components of one vector from those of the other.

Basis Vectors

Basis vectors are the building blocks of vectors in any space, serving as the reference points for vector components. In two dimensions, we commonly use \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) represents a unit vector along the x-axis and \( \mathbf{j} \) is a unit vector along the y-axis. They are fundamentally important because they provide a consistent frame of reference for measuring vector quantities.

In essence, any vector in the plane can be composed using a combination of multiples of these two basis vectors. When subtracting vectors, it is these multiples of the basis vectors that are individually subtracted, leading to a new vector's components that also align to the same underlying structure defined by \( \mathbf{i} \) and \( \mathbf{j} \) basis vectors.

Vector Operations

Vector operations, including addition, subtraction, and scalar multiplication, follow specific algebraic rules. Vector subtraction is particularly important as it allows one to determine the vector that spans from the tip of one vector to the tip of another. To perform vector subtraction, we take the vector to be subtracted and inverse its direction; technically, we multiply it by -1. Then it is added to the original vector. This operation boils down to subtracting the components of the second vector from the corresponding components of the first.

Following this approach, if we have two vectors \( \mathbf{A} \) and \( \mathbf{B} \) with their components already identified, vector subtraction is straightforward. We simply subtract the \( \mathbf{i} \) and \( \mathbf{j} \) components of \( \mathbf{B} \) from \( \mathbf{A} \), finding the resulting vector's components. This is demonstrated succinctly in the component-wise subtraction method, resulting in a new vector that represents the difference between \( \mathbf{A} \) and \( \mathbf{B} \) in terms of direction and magnitude.