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

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 \(\mathbf{a} = \langle 4, 0 \rangle\), we have: \[3 \mathbf{a} = 3 \cdot \langle 4, 0 \rangle = \langle 3 \times 4, 3 \times 0 \rangle = \langle 12, 0 \rangle.\]

Step 2: Add vectors a and b

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

Step 3: Subtract vectors b from a

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

Step 4: Calculate magnitude of a+b

To find \(\|\mathbf{a} + \mathbf{b}\|\), first find \(\mathbf{a} + \mathbf{b}\) which was \(\langle 4, -5 \rangle\) and then calculate its magnitude: \[\|\mathbf{a} + \mathbf{b}\| = \sqrt{4^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}.\]

Step 5: Calculate magnitude of a-b

To find \(\|\mathbf{a} - \mathbf{b}\|\), first find \(\mathbf{a} - \mathbf{b}\) which was \(\langle 4, 5 \rangle\) and then calculate its magnitude: \[\|\mathbf{a} - \mathbf{b}\| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}.\]

Key Concepts

Understanding Vector Addition

Vector addition is like calculating the result of two vectors pointing in different directions. Imagine using arrows, where the direction and length represent the vector's direction and magnitude. When you add vectors, you combine these arrows. This means summing up their respective components.
Given vectors - \(\mathbf{a} = \langle 4, 0 \rangle\) and - \(\mathbf{b} = \langle 0, -5 \rangle\),
the addition involves simply adding the horizontal components together, and the vertical components together.
So, \(\mathbf{a} + \mathbf{b} = \langle 4 + 0, 0 - 5 \rangle = \langle 4, -5 \rangle.\)
The result is a new vector that represents both the influence of \(\mathbf{a}\) and \(\mathbf{b}\). Think of this as finding a new path by stacking two movements one after the other.

Grasping Vector Subtraction

Vector subtraction is like taking the effect of one vector away from another. Visualization helps: consider two arrows representing vectors \( \mathbf{a} \) and \( \mathbf{b} \).
Subtracting \( \mathbf{b} \) from \( \mathbf{a} \) is akin to reversing the direction of \( \mathbf{b} \) and then adding it to \( \mathbf{a} \). This means subtracting each of \( \mathbf{b}'s \) components from the corresponding components of \( \mathbf{a} \).
For our vectors, \( \mathbf{a} = \langle 4, 0 \rangle \) and \( \mathbf{b} = \langle 0, -5 \rangle \), the subtraction works like this:
\[ \mathbf{a} - \mathbf{b} = \langle 4 - 0, 0 - (-5) \rangle = \langle 4 + 0, 0 + 5 \rangle = \langle 4, 5 \rangle. \]
Think of this as removing or cancelling the effects of \( \mathbf{b} \) from \( \mathbf{a} \).

Calculating the Magnitude of a Vector

The magnitude of a vector is its length, which is another word for the distance from its origin to its endpoint. It gives an idea of how "strong" a vector is in its space. To find the magnitude of any 2D vector \( \mathbf{v} = \langle x, y \rangle \), use Pythagoras' theorem:
\[ \|\mathbf{v}\| = \sqrt{x^2 + y^2}. \]
In our example, the magnitude of \( \mathbf{a} + \mathbf{b} \) is \( \mathbf{a} + \mathbf{b} = \langle 4, -5 \rangle \),
and its magnitude is
\[ \|\mathbf{a} + \mathbf{b}\| = \sqrt{4^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}. \]
Similarly, the magnitude of \( \mathbf{a} - \mathbf{b} \), which is \( \langle 4, 5 \rangle \), is also
\[ \|\mathbf{a} - \mathbf{b}\| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}. \]
This tells us exactly how long each resultant vector truly is in their respective coordinate systems.

Basics of Scalar Multiplication

Scalar multiplication involves taking a vector and stretching or compressing it. You multiply the vector by a simple number (a scalar), affecting both of its components proportionally. This changes the magnitude but not the direction.
Consider our vector \( \mathbf{a} = \langle 4, 0 \rangle \). When you multiply it by a scalar, like 3, each part of the vector gets multiplied by that scalar:
\[ 3\mathbf{a} = 3 \times \langle 4, 0 \rangle = \langle 3 \times 4, 3 \times 0 \rangle = \langle 12, 0 \rangle. \]
This operation has doubled the distance from the origin for every unit in the vector, effectively resizing the vector without changing its initial direction.