Problem 95 - 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 sum.

Step-by-Step Solution

Answer

The sum of two vectors, A and B, expressed in terms of \(\mathbf{i}\) and \(\mathbf{j}\), can be found by summing their corresponding \(\mathbf{i}\) and \(\mathbf{j}\) components separately. The resulting vector C is given by \(C = (a1 + b1)\mathbf{i} + (a2 + b2)\mathbf{j}\)

Step 1: Identify Vector Components

Identify the \(\mathbf{i}\) and \(\mathbf{j}\) components of each vector. For example, if vector A is represented as \(A = a1\mathbf{i} + a2\mathbf{j}\) and vector B as \(B = b1\mathbf{i} + b2\mathbf{j}\), then \(a1\), \(a2\), \(b1\), and \(b2\) are the \(\mathbf{i}\) and \(\mathbf{j}\) components of vectors A and B respectively.

Step 2: Sum the Corresponding Components

Sum the corresponding \(\mathbf{i}\) and \(\mathbf{j}\) components of both vectors. The \(\mathbf{i}\) component of the resulting vector (C) is the sum of the \(\mathbf{i}\) components of vector A and B (\(c1 = a1 + b1\)). Similarly, the \(\mathbf{j}\) component of vector C is the sum of the \(\mathbf{j}\) components of vector A and B (\(c2 = a2 + b2\)).

Step 3: Express the Result as a Vector

The resulting vector C is the sum of vector A and B and can be expressed in terms of \(\mathbf{i}\) and \(\mathbf{j}\). Therefore, the resulting vector C can be written as \(C = c1\mathbf{i} + c2\mathbf{j}\).\nThis is known as the component form of a vector. It precisely indicates the direction and magnitude of the vector relative to the x and y-axis.

Key Concepts

Vector Addition

Vector addition is a core concept in algebra that makes it possible to combine the magnitudes and directions of two different vectors. The process is a straightforward method used to determine the resultant vector which accurately represents the combined effect of the original vectors.

This operation follows the principle of the parallelogram law, which states that if two vectors are represented as adjacent sides of a parallelogram, their sum is given by the vector represented by the diagonal that passes through their common point.

In practical terms, when you have the vectors in component form, like \( A = a1\mathbf{i} + a2\mathbf{j} \) and \( B = b1\mathbf{i} + b2\mathbf{j} \), adding them simply means adding their corresponding \( \mathbf{i} \) and \( \mathbf{j} \) components. The resulting vector \( C \), which is the sum of \( A \) and \( B \), would have components \( c1 = a1 + b1 \) and \( c2 = a2 + b2 \).

Component Form of a Vector

The component form of a vector is a way of breaking down a vector into its individual parts, namely its horizontal and vertical components, which align with the x and y axes, respectively. Expressing vectors in this way allows for easier analysis and mathematical operations, like adding and subtracting vectors.

For instance, a vector represented by \( A = a1\mathbf{i} + a2\mathbf{j} \) explicitly shows how far along the x-axis (horizontal\(\mathbf{i}\) component) and y-axis (vertical \(\mathbf{j}\) component) the vector A extends. This form is integral when performing operations such as vector addition, as it enables a clear, component-wise combination of vectors.

Vector Components

Understanding vector components is essential because they provide information on the direction and magnitude of a vector in relation to the axes of a coordinate system. In two-dimensional space, every vector can be decomposed into two perpendicular components, typically aligned with the x-axis (horizontal) and y-axis (vertical).

These components are scalar quantities, which, when multiplied by the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) respectively, reconstitute the original vector. So, a vector \( A \) with a horizontal component of 3 and a vertical component of 4 would be written as \( A = 3\mathbf{i} + 4\mathbf{j} \).

By analyzing vector components, we can calculate the magnitude of a vector using Pythagoras' theorem and determine the direction by calculating the angle of the vector with respect to the horizontal axis.

i and j Unit Vectors

The unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) are fundamental to understanding vectors in algebra. They represent the basis vectors for the x-axis and y-axis, respectively, in two-dimensional space. Each unit vector has a magnitude of 1, which means they provide the purest representation of direction along their respective axes without influencing the magnitude of the vectors they are describing.

They are often used in expressing vectors in their component forms because any two-dimensional vector can be considered as a combination of these two unit vectors. To illustrate, a vector that stretches 5 units to the right (along the x-axis) and 2 units up (along the y-axis) would be represented as \( 5\mathbf{i} + 2\mathbf{j} \).

This notation not only simplifies calculations but also helps in visualizing vectors on a coordinate system, making it easier for students and professionals alike to work with vectors in various applications.