Problem 1 - 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 \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=2 \mathbf{i}+4 \mathbf{j}, \mathbf{b}=-\mathbf{i}+4 \mathbf{j}\)

Step-by-Step Solution

Answer

(a) \(6 \mathbf{i} + 12 \mathbf{j}\); (b) \(\mathbf{i} + 8 \mathbf{j}\); (c) \(3 \mathbf{i}\); (d) \(\sqrt{65}\); (e) 3.

Step 1: Calculate \(3 \mathbf{a}\)

To find \(3 \mathbf{a}\), we need to multiply each component of \(\mathbf{a} = 2 \mathbf{i} + 4 \mathbf{j}\) by 3. This gives us \(3 \cdot 2 \mathbf{i} + 3 \cdot 4 \mathbf{j} = 6 \mathbf{i} + 12 \mathbf{j}\).

Step 2: Add \(\mathbf{a} + \mathbf{b}\)

For addition, combine like components of \(\mathbf{a} = 2 \mathbf{i} + 4 \mathbf{j}\) and \(\mathbf{b} = -\mathbf{i} + 4 \mathbf{j}\). The result is \((2 + (-1)) \mathbf{i} + (4 + 4) \mathbf{j} = \mathbf{i} + 8 \mathbf{j}\).

Step 3: Subtract \(\mathbf{a} - \mathbf{b}\)

Subtract \(\mathbf{b} = -\mathbf{i} + 4 \mathbf{j}\) from \(\mathbf{a} = 2 \mathbf{i} + 4 \mathbf{j}\). The calculation is \((2 - (-1)) \mathbf{i} + (4 - 4) \mathbf{j} = 3 \mathbf{i}\).

Step 4: Calculate \(\|\mathbf{a} + \mathbf{b}\|\)

Previously, we found that \(\mathbf{a} + \mathbf{b} = \mathbf{i} + 8 \mathbf{j}\). The magnitude is calculated as \(\|\mathbf{a} + \mathbf{b}\| = \sqrt{1^2 + 8^2} = \sqrt{1 + 64} = \sqrt{65}\).

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

From Step 3, \(\mathbf{a} - \mathbf{b} = 3 \mathbf{i}\). The magnitude is \(\|3 \mathbf{i}\| = \sqrt{3^2} = \sqrt{9} = 3\).

Key Concepts

Vector Multiplication

Vector multiplication can involve different techniques, including scaling a vector by a scalar or performing a dot or cross product. In this context, we focus on scalar multiplication, which is the simplest form of vector multiplication.

If you have a vector \( \mathbf{a} = x \mathbf{i} + y \mathbf{j}\), and you multiply this vector by a scalar number, say 3, you will perform the operation on each component separately.

For instance:

• Multiply the scalar with each component: \(3 \cdot x \mathbf{i} + 3 \cdot y \mathbf{j}\).
• In the given exercise, \(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\), so \(3\mathbf{a} = 6\mathbf{i} + 12\mathbf{j}\).

This operation results in a new vector that points in the same direction as the original \(\mathbf{a}\), but is scaled in magnitude by three times the original length.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a new vector. This operation is performed component-wise.

Given two vectors,\(\mathbf{a} = x_1 \mathbf{i} + y_1 \mathbf{j}\) and\(\mathbf{b} = x_2 \mathbf{i} + y_2 \mathbf{j}\),their sum\(\mathbf{a} + \mathbf{b}\)is calculated as:

• Sum of the i-components:\((x_1 + x_2)\mathbf{i}\)
• Sum of the j-components:\((y_1 + y_2)\mathbf{j}\)

In the given problem, adding\(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\)and\(\mathbf{b} = -\mathbf{i} + 4\mathbf{j} \),the resulting vector is\((2 + (-1))\mathbf{i} + (4 + 4)\mathbf{j} = \mathbf{i} + 8\mathbf{j}\).
Vector addition is like adding arrows tip-to-tail, reflecting the net effect of both vectors.

Vector Subtraction

Vector subtraction is similar to vector addition but involves reversing the direction of the second vector before adding the components.

If you subtract vector \(\mathbf{b} = x_2 \mathbf{i} + y_2 \mathbf{j}\)from \(\mathbf{a} = x_1 \mathbf{i} + y_1 \mathbf{j}\),the result is a new vector:

• For i-component:\((x_1 - x_2)\mathbf{i}\)
• For j-component:\((y_1 - y_2)\mathbf{j}\)

In this exercise, \(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\), and \(\mathbf{b} = -\mathbf{i} + 4\mathbf{j}\),giving us \(\mathbf{a} - \mathbf{b} = 3\mathbf{i}\).
Essentially, you find the difference by 'adding' the opposite of \(\mathbf{b}\), which essentially cancels out components if they are equal.

Vector Magnitude

The magnitude of a vector, also known as its length or size, shows how far it stretches in the space.

To find the magnitude of a vector\(\mathbf{v} = x\mathbf{i} + y\mathbf{j}\),use the formula:

• \(\|\mathbf{v}\| = \sqrt{x^2 + y^2}\)

This calculation gives a scalar (a single number) representing the vector's size.

For instance:

• If\(\mathbf{v} = \mathbf{i} + 8\mathbf{j}\), \(\|\mathbf{v}\| = \sqrt{1^2 + 8^2} = \sqrt{65}\).
• For\(\mathbf{a} - \mathbf{b} = 3\mathbf{i}\), the magnitude is \(\| 3\mathbf{i}\| = 3\).

The vector magnitude tells you how long a vector is, without regard to its direction.