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

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 3a

To find \(3 \mathbf{a}\), multiply each component of \(\mathbf{a}\) by 3. Given \(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\), then:\[ 3\mathbf{a} = 3(2\mathbf{i} + 4\mathbf{j}) = 6\mathbf{i} + 12\mathbf{j} \]

Step 2: Calculate a + b

Add the corresponding components of \(\mathbf{a}\) and \(\mathbf{b}\). Given \(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\) and \(\mathbf{b} = -\mathbf{i} + 4\mathbf{j}\), then:\[ \mathbf{a} + \mathbf{b} = (2\mathbf{i} + 4\mathbf{j}) + (-\mathbf{i} + 4\mathbf{j}) = (2 - 1)\mathbf{i} + (4 + 4)\mathbf{j} = \mathbf{i} + 8\mathbf{j} \]

Step 3: Calculate a - b

Subtract the corresponding components of \(\mathbf{b}\) from \(\mathbf{a}\). Given \(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\) and \(\mathbf{b} = -\mathbf{i} + 4\mathbf{j}\), then:\[ \mathbf{a} - \mathbf{b} = (2\mathbf{i} + 4\mathbf{j}) - (-\mathbf{i} + 4\mathbf{j}) = (2 + 1)\mathbf{i} + (4 - 4)\mathbf{j} = 3\mathbf{i} \]

Step 4: Calculate ||a + b||

Find the magnitude of \(\mathbf{a} + \mathbf{b}\). We already calculated \(\mathbf{a} + \mathbf{b} = \mathbf{i} + 8\mathbf{j}\). The magnitude is:\[ \| \mathbf{a} + \mathbf{b} \| = \sqrt{(1)^2 + (8)^2} = \sqrt{1 + 64} = \sqrt{65} \]

Step 5: Calculate ||a - b||

Find the magnitude of \(\mathbf{a} - \mathbf{b}\). We have \(\mathbf{a} - \mathbf{b} = 3\mathbf{i}\). The magnitude is:\[ \| \mathbf{a} - \mathbf{b} \| = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3 \]

Key Concepts

vector addition

Adding vectors is like combining two journeys taken by a person. You take the first trip path and add it to the second trip path. In the world of vectors, this means adding each corresponding component of the vectors together. Given

• \( \mathbf{a} = 2\mathbf{i} + 4\mathbf{j} \)
• \( \mathbf{b} = -\mathbf{i} + 4\mathbf{j} \)

Adding them works by adding the i-components from both vectors, and then the j-components separately:

• \( (2 + (-1))\mathbf{i} \) = \( \mathbf{i} \)
• \( (4 + 4)\mathbf{j} \) = \( 8\mathbf{j} \)

This results in the vector \( \mathbf{a} + \mathbf{b} = \mathbf{i} + 8\mathbf{j} \).
Remember to always keep i's with i's and j's with j's during addition of vectors.

vector subtraction

Vector subtraction is like finding the difference in paths between two journeys. For this, we take each component of one vector and subtract the corresponding component of the other vector. So, for vectors

• \( \mathbf{a} = 2\mathbf{i} + 4\mathbf{j} \)
• \( \mathbf{b} = -\mathbf{i} + 4\mathbf{j} \)

We subtract i-components and j-components separately:

• \( (2 - (-1))\mathbf{i} = 3\mathbf{i} \)
• \( (4 - 4)\mathbf{j} = 0\)

This gives us the vector \( \mathbf{a} - \mathbf{b} = 3\mathbf{i} \).Subtraction results in a vector representing the difference between the two original vectors.
Be sure to handle the signs carefully to avoid mistakes.

vector magnitude

The magnitude of a vector measures its length. Imagine it as the actual distance, without direction, of the path represented by the vector.
To find the magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \), you use the formula:\[ \| \mathbf{v} \| = \sqrt{a^2 + b^2} \]For example, if you want to find the magnitude of \( \mathbf{a} + \mathbf{b} = \mathbf{i} + 8\mathbf{j} \):\[\| \mathbf{a} + \mathbf{b} \| = \sqrt{(1)^2 + (8)^2} = \sqrt{1 + 64} = \sqrt{65} \]For \( \mathbf{a} - \mathbf{b} = 3\mathbf{i} \), the magnitude calculation is:\[\| \mathbf{a} - \mathbf{b} \| = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3 \]The magnitude essentially transforms your vector into a single positive number, indicating its size.

scalar multiplication

Scalar multiplication involves taking a vector and making it longer or shorter. You multiply every component of the vector by the same number, called a scalar. For instance, if you have the vector \( \mathbf{a} = 2\mathbf{i} + 4\mathbf{j} \),and you want to multiply it by 3, each component must be multiplied by 3:

• \( 3 \times 2\mathbf{i} = 6\mathbf{i} \)
• \( 3 \times 4 \mathbf{j} = 12\mathbf{j} \)

This results in a new vector \( 3\mathbf{a} = 6\mathbf{i} + 12\mathbf{j} \). Scalar multiplication scales the vector, keeping its direction consistent.
This mechanism can either stretch or shrink a vector depending on whether the scalar is greater than or less than one.