Problem 11 - Advanced Engineering Mathematics

Question

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\)

Step-by-Step Solution

Answer

(a) \( 10\mathbf{i} - 12\mathbf{j} \), (b) \( 12\mathbf{i} - 17\mathbf{j} \).

Step 1: Understand the Problem

We have two vectors: \( \mathbf{a} = \mathbf{i} - \mathbf{j} \) and \( \mathbf{b} = -3\mathbf{i} + 4\mathbf{j} \). We need to find the resulting vectors from the expressions \( 4\mathbf{a} - 2\mathbf{b} \) and \( -3\mathbf{a} - 5\mathbf{b} \).

Step 2: Compute \( 4\mathbf{a} \)

Calculate \( 4\mathbf{a} \) by multiplying each component of \( \mathbf{a} \) by 4: \( 4\mathbf{a} = 4(\mathbf{i} - \mathbf{j}) = 4\mathbf{i} - 4\mathbf{j} \).

Step 3: Compute \(-2\mathbf{b}\)

Calculate \(-2\mathbf{b}\) by multiplying each component of \( \mathbf{b} \) by \(-2\): \(-2\mathbf{b} = -2(-3\mathbf{i} + 4\mathbf{j}) = 6\mathbf{i} - 8\mathbf{j} \).

Step 4: Calculate \( 4\mathbf{a} - 2\mathbf{b} \)

Add the results from Step 2 and Step 3: \( 4\mathbf{a} - 2\mathbf{b} = (4\mathbf{i} - 4\mathbf{j}) + (6\mathbf{i} - 8\mathbf{j}) = 10\mathbf{i} - 12\mathbf{j} \).

Step 5: Compute \(-3\mathbf{a} \)

Calculate \(-3\mathbf{a} \) by multiplying each component of \( \mathbf{a} \) by \(-3\): \(-3\mathbf{a} = -3(\mathbf{i} - \mathbf{j}) = -3\mathbf{i} + 3\mathbf{j} \).

Step 6: Compute \(-5\mathbf{b}\)

Calculate \(-5\mathbf{b}\) by multiplying each component of \( \mathbf{b} \) by \(-5\): \(-5\mathbf{b} = -5(-3\mathbf{i} + 4\mathbf{j}) = 15\mathbf{i} - 20\mathbf{j} \).

Step 7: Calculate \(-3\mathbf{a} - 5\mathbf{b}\)

Add the results from Step 5 and Step 6: \(-3\mathbf{a} - 5\mathbf{b} = (-3\mathbf{i} + 3\mathbf{j}) + (15\mathbf{i} - 20\mathbf{j}) = 12\mathbf{i} - 17\mathbf{j} \).

Key Concepts

Vector Operations

Vector operations are mathematical expressions that involve vectors, which are quantities having both magnitude and direction. These operations include addition, multiplication by scalars, and subtraction. With these operations, vectors can be manipulated and transformed to solve a variety of problems in physics, engineering, and computer science.
Understanding vector operations is crucial as they are the foundation for more complex concepts like vector calculus and linear algebra.
There are two types of vector multiplication: dot product and cross product, but for the exercise in consideration, we focus on scalar multiplication. **Scalar multiplication** involves multiplying a vector by a real number (scalar). This changes the magnitude of the vector without affecting its direction.
By familiarizing yourself with vector operations, you can easily tackle problems involving forces, velocity, and acceleration in a more systematic manner. You will often encounter vector operations in physics problems where multiple forces are involved.

Vector Multiplication

When we discuss vector multiplication in the context of this exercise, we specifically mean scalar multiplication. Scalar multiplication involves multiplying each component of a vector by a scalar.
For example, given the vector \( \mathbf{a} = \mathbf{i} - \mathbf{j} \) and the scalar 4, scalar multiplication is applied as follows:

• Multiply the \( \mathbf{i} \) component by 4 to get \( 4\mathbf{i} \).
• Multiply the \( \mathbf{j} \) component by 4 to get \( -4\mathbf{j} \).

Thus, the resulting vector after scalar multiplication is \( 4\mathbf{a} = 4\mathbf{i} - 4\mathbf{j} \).
Scalar multiplication is used to change the size of vectors without altering their direction. It's similar to stretching or shrinking the vector in the same direction it originally pointed. Similarly, negative scalars will also reverse the vector's direction. This is crucial in understanding problems that involve scaling forces or velocities in physics.

Vector Addition

Vector addition entails combining two or more vectors to obtain a resultant vector. When adding vectors, we add their respective components. This operation takes into account both the direction and magnitude of the vectors involved.
Let's examine the operation using vectors \( \mathbf{a} = \mathbf{i} - \mathbf{j} \) and \( \mathbf{b} = -3\mathbf{i} + 4\mathbf{j} \).
Suppose we want to add vectors \( 4\mathbf{a} \) and \(-2\mathbf{b}\). First, compute each expression separately:

• For \( 4\mathbf{a} = 4\mathbf{i} - 4\mathbf{j} \), the components are \( 4\mathbf{i} \) and \( -4\mathbf{j} \).
• For \(-2\mathbf{b} = 6\mathbf{i} - 8\mathbf{j} \), the components are \( 6\mathbf{i} \) and \( -8\mathbf{j} \).

Then, add the corresponding components:

• Add the \(\mathbf{i}\) components: \(4\mathbf{i} + 6\mathbf{i} = 10\mathbf{i}\).
• Add the \(\mathbf{j}\) components: \(-4\mathbf{j} - 8\mathbf{j} = -12\mathbf{j}\).

This results in the vector \( 10\mathbf{i} - 12\mathbf{j} \), demonstrating how vector addition works.
Vector addition is crucial in problem-solving, especially in physics, where it helps understand resultant forces and direction when multiple forces act on a body.