Problem 3 - 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) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=\langle 4,0\rangle, \mathbf{b}=\langle 0,-5\rangle $$

Step-by-Step Solution

Answer

(a) \(\langle 12, 0 \rangle\), (b) \(\langle 4, -5 \rangle\), (c) \(\langle 4, 5 \rangle\), (d) \(\sqrt{41}\), (e) \(\sqrt{41}\).

Step 1: Calculate 3a

To find \(3 \mathbf{a}\), multiply each component of \(\mathbf{a}\) by 3. Given \(\mathbf{a} = \langle 4,0 \rangle\), \(3 \mathbf{a} = \langle 3 \times 4, 3 \times 0 \rangle = \langle 12, 0 \rangle\).

Step 2: Add vectors a and b

To find \(\mathbf{a}+\mathbf{b}\), add the corresponding components of \(\mathbf{a}\) and \(\mathbf{b}\). Thus, \(\mathbf{a} + \mathbf{b} = \langle 4, 0 \rangle + \langle 0, -5 \rangle = \langle 4 + 0, 0 - 5 \rangle = \langle 4, -5 \rangle\).

Step 3: Subtract vector b from a

To find \(\mathbf{a}-\mathbf{b}\), subtract the components of \(\mathbf{b}\) from \(\mathbf{a}\). Thus, \(\mathbf{a} - \mathbf{b} = \langle 4, 0 \rangle - \langle 0, -5 \rangle = \langle 4 - 0, 0 + 5 \rangle = \langle 4, 5 \rangle\).

Step 4: Magnitude of a + b

To find \(\|\mathbf{a}+\mathbf{b}\|\), calculate the magnitude of \(\mathbf{a}+\mathbf{b}\). The magnitude formula is \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2}\). Therefore, \(\|\mathbf{a}+\mathbf{b}\| = \sqrt{4^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}\).

Step 5: Magnitude of a - b

To find \(\|\mathbf{a}-\mathbf{b}\|\), calculate the magnitude of \(\mathbf{a}-\mathbf{b}\). Thus, \(\|\mathbf{a}-\mathbf{b}\| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}\).

Key Concepts

Vector Addition

Understanding vector addition is essential for anyone studying physics, engineering, or any field involving multi-component forces. In vector addition, you combine two or more vectors to find the resultant vector. Each vector is represented by its components along the coordinate axes.
To add vectors, simply add their corresponding components. For example, if you have vectors \( \mathbf{a} = \langle 4, 0 \rangle \) and \( \mathbf{b} = \langle 0, -5 \rangle \), the result of their addition is:
\[ \mathbf{a} + \mathbf{b} = \langle 4 + 0, 0 - 5 \rangle = \langle 4, -5 \rangle \]
This simple operation is foundational, as it allows vectors to be added graphically by placing them head to tail, resulting in the same magnitude and direction as the arithmetic approach.

Vector Subtraction

Vector subtraction might seem a bit tricky at first, but it's just as straightforward as addition once you understand it. The key is realizing that subtracting a vector is equivalent to adding its negative.
Let's say you're tasked with \( \mathbf{a} = \langle 4, 0 \rangle \) and \( \mathbf{b} = \langle 0, -5 \rangle \). To find \( \mathbf{a} - \mathbf{b} \), subtract each corresponding component of \( \mathbf{b} \) from \( \mathbf{a} \):
\[ \mathbf{a} - \mathbf{b} = \langle 4 - 0, 0 - (-5) \rangle = \langle 4, 0 + 5 \rangle = \langle 4, 5 \rangle \]
Notice how the subtraction converted into addition when applying to the negative component, changing the direction of the vector.

Scalar Multiplication

Scalar multiplication is a basic operation that scales a vector by a certain amount. This process involves multiplying each component of a vector by the scalar (a real number).
For instance, if given a vector \( \mathbf{a} = \langle 4, 0 \rangle \) and asked to compute \( 3 \mathbf{a} \), multiply each component of \( \mathbf{a} \) by 3:
\[ 3 \cdot \langle 4, 0 \rangle = \langle 3 \times 4, 3 \times 0 \rangle = \langle 12, 0 \rangle \]
Scalar multiplication changes the magnitude of the vector but not its direction. However, if the scalar is negative, the direction of the resulting vector is opposite to that of the original vector.

Magnitude of a Vector

To determine the magnitude of a vector, which represents its length, you apply the Pythagorean theorem. This is simply the square root of the sum of the squares of its components.
For a vector \( \mathbf{x} = \langle x_1, x_2 \rangle \), the magnitude \( \| \mathbf{x} \| \) is calculated as:
\[ \| \mathbf{x} \| = \sqrt{x_1^2 + x_2^2} \]
If we consider finding the magnitude of the sum of two vectors, such as \( \mathbf{a} + \mathbf{b} = \langle 4, -5 \rangle \), the calculation becomes:
\[ \| \mathbf{a} + \mathbf{b} \| = \sqrt{4^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41} \]
This formula is fundamental in physics for determining the strength of forces and understanding vector quantities in space.