Problem 5 - Advanced Engineering Mathematics

Question

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

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) \(3\sqrt{10}\), (e) \(\sqrt{34}\)

Step 1: Calculate 3a

Given the vector \( \mathbf{a} = -3 \mathbf{i} + 2 \mathbf{j} \), we need to find \( 3 \mathbf{a} \).Multiply each component of \( \mathbf{a} \) by 3:\[3\mathbf{a} = 3(-3 \mathbf{i} + 2 \mathbf{j}) = -9 \mathbf{i} + 6 \mathbf{j}\]Thus, \( 3\mathbf{a} = -9 \mathbf{i} + 6 \mathbf{j} \).

Step 2: Calculate a + b

Given \( \mathbf{a} = -3 \mathbf{i} + 2 \mathbf{j} \) and \( \mathbf{b} = 7 \mathbf{j} \), we need to find \( \mathbf{a} + \mathbf{b} \).Add corresponding components:\[\mathbf{a} + \mathbf{b} = (-3 \mathbf{i} + 2 \mathbf{j}) + (0 \mathbf{i} + 7 \mathbf{j}) = -3 \mathbf{i} + (2 + 7) \mathbf{j} = -3 \mathbf{i} + 9 \mathbf{j}\]Thus, \( \mathbf{a} + \mathbf{b} = -3 \mathbf{i} + 9 \mathbf{j} \).

Step 3: Calculate a - b

Given \( \mathbf{a} = -3 \mathbf{i} + 2 \mathbf{j} \) and \( \mathbf{b} = 7 \mathbf{j} \), we need to find \( \mathbf{a} - \mathbf{b} \).Subtract corresponding components:\[\mathbf{a} - \mathbf{b} = (-3 \mathbf{i} + 2 \mathbf{j}) - (0 \mathbf{i} + 7 \mathbf{j}) = -3 \mathbf{i} + (2 - 7) \mathbf{j} = -3 \mathbf{i} - 5 \mathbf{j}\]Thus, \( \mathbf{a} - \mathbf{b} = -3 \mathbf{i} - 5 \mathbf{j} \).

Step 4: Calculate the magnitude of a + b

To find \( \|\mathbf{a} + \mathbf{b}\| \), first use the result from Step 2, which is \( \mathbf{a} + \mathbf{b} = -3 \mathbf{i} + 9 \mathbf{j} \).Calculate the magnitude:\[\|\mathbf{a} + \mathbf{b}\| = \sqrt{(-3)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}\]Thus, \( \|\mathbf{a} + \mathbf{b}\| = 3\sqrt{10} \).

Step 5: Calculate the magnitude of a - b

To find \( \|\mathbf{a} - \mathbf{b}\| \), first use the result from Step 3, which is \( \mathbf{a} - \mathbf{b} = -3 \mathbf{i} - 5 \mathbf{j} \).Calculate the magnitude:\[\|\mathbf{a} - \mathbf{b}\| = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}\]Thus, \( \|\mathbf{a} - \mathbf{b}\| = \sqrt{34} \).

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to form a single vector. When adding vectors, you sum up their corresponding components independently.
For example, if you have vectors \( \mathbf{a} = -3 \mathbf{i} + 2 \mathbf{j} \) and \( \mathbf{b} = 0 \mathbf{i} + 7 \mathbf{j} \), you would add them by performing the following steps:

• Add the \("i"\) components: \(-3 + 0 = -3\)
• Add the \("j"\) components: \(2 + 7 = 9\)

This results in the vector \( \mathbf{a} + \mathbf{b} = -3 \mathbf{i} + 9 \mathbf{j} \).
Visualize vector addition as putting vectors tip-to-tail and drawing the resulting vector from the start of the first to the end of the last.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors. This is done by subtracting their corresponding components.
Using the vectors \( \mathbf{a} = -3 \mathbf{i} + 2 \mathbf{j} \) and \( \mathbf{b} = 0 \mathbf{i} + 7 \mathbf{j} \), vector subtraction looks like this:

• Subtract the \("i"\) components: \(-3 - 0 = -3\)
• Subtract the \("j"\) components: \(2 - 7 = -5\)

This results in the vector \( \mathbf{a} - \mathbf{b} = -3 \mathbf{i} - 5 \mathbf{j} \).
You can think of vector subtraction as adding a vector and its negative. Place the vector opposite in direction and add tip-to-tail.

Magnitude of a Vector

The magnitude of a vector quantifies its length without considering the direction. It's crucial in finding the length of the vector in a space.
For a vector given by \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \), the magnitude \( \|\mathbf{v}\| \) is calculated using the formula: \[ \|\mathbf{v}\| = \sqrt{a^2 + b^2} \]
For example, the magnitude of \( \mathbf{a} + \mathbf{b} = -3 \mathbf{i} + 9 \mathbf{j} \) is found using: \[ \|-3 \mathbf{i} + 9 \mathbf{j}\| = \sqrt{(-3)^2 + 9^2} = \sqrt{90} = 3\sqrt{10} \]
Similarly, for \( \mathbf{a} - \mathbf{b} = -3 \mathbf{i} - 5 \mathbf{j} \), the magnitude is: \[ \|-3 \mathbf{i} - 5 \mathbf{j}\| = \sqrt{(-3)^2 + (-5)^2} = \sqrt{34} \]
Magnitude is always a non-negative number and represents the vector's length.

Scalar Multiplication of Vectors

Scalar multiplication involves scaling a vector by a constant, known as a scalar. This operation adjusts the magnitude of the vector while preserving its direction.
To perform scalar multiplication, multiply each component of the vector by the scalar. For vector \( \mathbf{a} = -3 \mathbf{i} + 2 \mathbf{j} \) and scalar 3, you perform the multiplication as follows:

• Multiply the \("i"\) component: \(3 \times (-3) = -9\)
• Multiply the \("j"\) component: \(3 \times 2 = 6\)

This results in the vector \( 3\mathbf{a} = -9 \mathbf{i} + 6 \mathbf{j} \).
Scalar multiplication affects the vector's length proportionally to the scalar value, with direction staying constant unless multiplied by a negative scalar, which reverses direction.