Problem 5 - Advanced Engineering Mathematics

Question

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}, \mathbf{b}=7 \mathbf{j} $$

Step-by-Step Solution

Answer

(a) \(-9\mathbf{i}+6\mathbf{j}\), (b) \(-3\mathbf{i}+9\mathbf{j}\), (c) \(-3\mathbf{i}-5\mathbf{j}\), (d) \(\sqrt{90}\), (e) \(\sqrt{34}\)

Step 1: Compute 3a

To find \(3\mathbf{a}\), multiply each component of \(\mathbf{a} = -3\mathbf{i} + 2\mathbf{j}\) by 3. This results in \(3(-3)\mathbf{i} + 3(2)\mathbf{j} = -9\mathbf{i} + 6\mathbf{j}\).

Step 2: Compute a + b

To find \(\mathbf{a} + \mathbf{b}\), sum up the components of \(\mathbf{a} = -3\mathbf{i} + 2\mathbf{j}\) and \(\mathbf{b} = 7\mathbf{j}\). The \(\mathbf{i}\) component is \(-3\) and the \(\mathbf{j}\) components add up to \(2+7\). Thus, \(\mathbf{a} + \mathbf{b} = -3\mathbf{i} + 9\mathbf{j}\).

Step 3: Compute a - b

To find \(\mathbf{a} - \mathbf{b}\), subtract the components of \(\mathbf{b} = 7\mathbf{j}\) from \(\mathbf{a} = -3\mathbf{i} + 2\mathbf{j}\). The \(\mathbf{i}\) component stays \(-3\) and the \(\mathbf{j}\) component is \(2 - 7\). Thus, \(\mathbf{a} - \mathbf{b} = -3\mathbf{i} - 5\mathbf{j}\).

Step 4: Compute the Magnitude of a + b

To find \(\|\mathbf{a} + \mathbf{b}\|\), calculate the magnitude of \(\mathbf{a} + \mathbf{b} = -3\mathbf{i} + 9\mathbf{j}\). The formula for the magnitude is \(\sqrt{(-3)^2 + 9^2}\). Simplifying, \(\|\mathbf{a} + \mathbf{b}\| = \sqrt{9 + 81} = \sqrt{90}\).

Step 5: Compute the Magnitude of a - b

To find \(\|\mathbf{a} - \mathbf{b}\|\), calculate the magnitude of \(\mathbf{a} - \mathbf{b} = -3\mathbf{i} - 5\mathbf{j}\). The formula for the magnitude is \(\sqrt{(-3)^2 + (-5)^2}\). Simplifying, \(\|\mathbf{a} - \mathbf{b}\| = \sqrt{9 + 25} = \sqrt{34}\).

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to form a new vector. This is done by adding together the corresponding components of the vectors involved.
For example, consider vectors \( \mathbf{a} = -3\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{b} = 7\mathbf{j} \). To add these vectors, simply add the \( \mathbf{i} \) and \( \mathbf{j} \) components separately.

• The \( \mathbf{i} \) component of \( \mathbf{a} + \mathbf{b} \) is \(-3 + 0 = -3\).
• The \( \mathbf{j} \) component of \( \mathbf{a} + \mathbf{b} \) is \(2 + 7 = 9\).

Thus, the resulting vector from this addition is \( \mathbf{a} + \mathbf{b} = -3\mathbf{i} + 9\mathbf{j} \). Vector addition is widely used in physics and engineering to calculate resultant forces or velocities.

Vector Subtraction

Vector subtraction is similar to vector addition, but involves subtracting the corresponding components of two vectors to find the difference.
When tasked with finding \( \mathbf{a} - \mathbf{b} \) for vectors \( \mathbf{a} = -3\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{b} = 7\mathbf{j} \), follow these steps:

• Subtract the \( \mathbf{i} \) components: \(-3 - 0 = -3\).
• Subtract the \( \mathbf{j} \) components: \(2 - 7 = -5\).

Therefore, \( \mathbf{a} - \mathbf{b} = -3\mathbf{i} - 5\mathbf{j} \).
Vector subtraction is useful in computing the relative position or differences in velocities between objects.

Magnitude of a Vector

The magnitude of a vector provides a measure of its length. It is calculated using the square root of the sum of the squares of its components.
The formula for the magnitude \( \|\mathbf{v}\| \) of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) is:\[\|\mathbf{v}\| = \sqrt{a^2 + b^2}\]For \( \mathbf{a} + \mathbf{b} \equiv -3\mathbf{i} + 9\mathbf{j} \), the magnitude is:\[\|\mathbf{a} + \mathbf{b}\| = \sqrt{(-3)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90}\]For \( \mathbf{a} - \mathbf{b} \equiv -3\mathbf{i} - 5\mathbf{j} \), the magnitude is:\[\|\mathbf{a} - \mathbf{b}\| = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}\]Understanding the magnitude helps one determine the size of a vector, which is crucial in both theoretical and practical applications like determining distances or speeds.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a real number, called a scalar, which scales the vector by that amount.
Consider multiplying vector \( \mathbf{a} = -3\mathbf{i} + 2\mathbf{j} \) by the scalar 3. Each component of the vector is scaled:

• Multiply the \( \mathbf{i} \) component: \(3 \times (-3) = -9\).
• Multiply the \( \mathbf{j} \) component: \(3 \times 2 = 6\).

The result of this scalar multiplication is \( 3\mathbf{a} = -9\mathbf{i} + 6\mathbf{j} \).
Scalar multiplication is key when adjusting the magnitude of vectors without changing their direction.