Problem 22 - Advanced Engineering Mathematics

Question

Determine a scalar \(c\) so that \(\mathbf{a}=3 \mathbf{i}+c \mathbf{j}\) and \(\mathbf{b}=-\mathbf{i}+9 \mathbf{j}\) are parallel.

Step-by-Step Solution

Answer

The scalar \(c\) is \(-27\).

Step 1: Understanding Parallel Vectors

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel if one is a scalar multiple of the other. This means \(\mathbf{a} = k\mathbf{b}\) or \(\mathbf{b} = k\mathbf{a}\) for some scalar \(k\).

Step 2: Equating the Vectors

Given that \(\mathbf{a} = 3 \mathbf{i} + c \mathbf{j}\) and \(\mathbf{b} = -\mathbf{i} + 9 \mathbf{j}\), we express \(\mathbf{a}\) as a scalar multiple of \(\mathbf{b}\) by setting \(\mathbf{a} = k\mathbf{b}\):\[3 \mathbf{i} + c \mathbf{j} = k(-\mathbf{i} + 9 \mathbf{j})\]

Step 3: Solving for the Scalar k

From the equation, we equate the components separately.For the \(\mathbf{i}\) component:\[3 = -k\]Solving for \(k\), we get \(k = -3\).For the \(\mathbf{j}\) component:\[c = 9k\]

Step 4: Solving for Scalar c

Substitute \(k = -3\) into the \(\mathbf{j}\) component equation:\[c = 9(-3)\]\[c = -27\]

Step 5: Verifying the Solution

Check if substituting \(c = -27\) satisfies the parallel condition:\[-\mathbf{i} + 9 \mathbf{j}\] times \(-3\) gives \[3 \mathbf{i} - 27 \mathbf{j}\], which matches \(\mathbf{a}\). Therefore, the vectors are parallel.

Key Concepts

Vector Algebra

Vector algebra is a branch of mathematics that deals with essential operations on vectors, which are quantities having both magnitude and direction. Vectors are often represented in terms of their components, typically in the form of a combination of the basis unit vectors, such as \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) in three-dimensional space.
Here, the vector \( \mathbf{a} = 3 \mathbf{i} + c \mathbf{j} \) represents a vector in the two-dimensional plane, with 3 and \( c \) as its components along the \( x \)-axis and \( y \)-axis, respectively. Similarly, \( \mathbf{b} = -\mathbf{i} + 9 \mathbf{j} \), another two-dimensional vector, has components -1 and 9 along the \( x \)-axis and \( y \)-axis.
In vector algebra, understanding how to manipulate these vectors using operations such as addition, subtraction, and scalar multiplication is crucial, especially when solving problems involving the parallelism and equality of vectors.

Scalar Multiple

A scalar multiple involves multiplying a vector by a scalar (a real number) to produce another vector. This operation changes the magnitude of the original vector while keeping its direction the same, provided the scalar is positive. If the scalar is negative, it reverses the direction of the vector.
Understanding scalar multiples is key to determining when two vectors are parallel. Two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), can be expressed as \( \mathbf{a} = k \mathbf{b} \) or \( \mathbf{b} = k \mathbf{a} \) if they are parallel, where \( k \) represents our scalar multiple. In this exercise, we are required to find a scalar \( k \) such that the vectors \( \mathbf{a} = 3 \mathbf{i} + c \mathbf{j} \) and \( \mathbf{b} = -\mathbf{i} + 9 \mathbf{j} \) align perfectly in direction.
By carrying out these operations, specifically distinguishing each component separately, we're able to find the appropriate scalar that ensures the vectors are parallel, enhancing our understanding of how vectors can be manipulated in practical applications.

Component-wise Comparison

Component-wise comparison involves comparing each element of vector components individually. For vectors \( \mathbf{a} = 3 \mathbf{i} + c \mathbf{j} \) and \( \mathbf{b} = -\mathbf{i} + 9 \mathbf{j} \), we break down the relationship \( \mathbf{a} = k \mathbf{b} \) by looking at the \( \mathbf{i} \) and \( \mathbf{j} \) components separately.
For the \( \mathbf{i} \) component, we set up the equation:
- \( 3 = -k \)
Solving this, we discover that \( k = -3 \). This helps tie the respective \( \mathbf{i} \)-components of each vector together.
Then, for the \( \mathbf{j} \) component, the equation is:
- \( c = 9k \)
Substituting \( k = -3 \) into it, we get \( c = -27 \). This reveals the specific \( c \) value that aligns \( \mathbf{a} \) in direction with \( \mathbf{b} \), thus making them parallel.
By examining components separately, this method solidly confirms the solution with clarity and precision, ensuring each part of the vector satisfies the parallel condition.

Solution Verification

Verifying the solution is a step that ensures the calculated results truly satisfy the conditions given in the problem. Verification helps confirm accuracy and reliability of the solution derived from analytical methods.
After determining \( c = -27 \), we substitute it back into the vector \( \mathbf{a} = 3 \mathbf{i} - 27 \mathbf{j} \). Consequently, for \( \mathbf{b} = -\mathbf{i} + 9 \mathbf{j} \), multiplying by \( k = -3 \) results in:
- \( -3(-\mathbf{i} + 9 \mathbf{j}) = 3 \mathbf{i} - 27 \mathbf{j} \).
This not only matches \( \mathbf{a} \) directly but also confirms that the solution meets the parallelism condition.
Thus, verifying the calculated results corroborates that our mathematical approaches align with the prerequisites of vector parallelism and provide a robust check on the methods used in component-wise analysis.