Problem 6 - 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) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=\langle 1,3\rangle, \mathbf{b}=-5 \mathbf{a} $$

Step-by-Step Solution

Answer

3a = ⟨3, 9⟩; a+b = ⟨-4, -12⟩; a-b = ⟨6, 18⟩; |a+b| = 4√10; |a-b| = 6√10.

Step 1: Find Vector b

Vector \( \mathbf{b} \) is given as \( -5 \mathbf{a} \). So, calculate \( \mathbf{b} \) by multiplying each component of \( \mathbf{a} = \langle 1, 3 \rangle \) by \(-5\).\(\mathbf{b} = -5 \times \langle 1, 3 \rangle = \langle -5 \times 1, -5 \times 3 \rangle = \langle -5, -15 \rangle.\)

Step 2: Calculate 3a

To find \(3 \mathbf{a}\), multiply each component of \( \mathbf{a} = \langle 1, 3 \rangle \) by 3.\(3 \mathbf{a} = \langle 3 \times 1, 3 \times 3 \rangle = \langle 3, 9 \rangle.\)

Step 3: Calculate a + b

Add the corresponding components of \( \mathbf{a} = \langle 1, 3 \rangle \) and \( \mathbf{b} = \langle -5, -15 \rangle \).\(\mathbf{a} + \mathbf{b} = \langle 1 + (-5), 3 + (-15) \rangle = \langle -4, -12 \rangle.\)

Step 4: Calculate a - b

Subtract the components of \( \mathbf{b} = \langle -5, -15 \rangle \) from \( \mathbf{a} = \langle 1, 3 \rangle \).\(\mathbf{a} - \mathbf{b} = \langle 1 - (-5), 3 - (-15) \rangle = \langle 6, 18 \rangle.\)

Step 5: Calculate Magnitude of a + b

Use the formula for the magnitude of a vector: \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \). Apply it to \( \mathbf{a} + \mathbf{b} = \langle -4, -12 \rangle \).\(\| \mathbf{a} + \mathbf{b} \| = \sqrt{(-4)^2 + (-12)^2} = \sqrt{16 + 144} = \sqrt{160} = 4 \sqrt{10}.\)

Step 6: Calculate Magnitude of a - b

Apply the magnitude formula to \( \mathbf{a} - \mathbf{b} = \langle 6, 18 \rangle \).\(\| \mathbf{a} - \mathbf{b} \| = \sqrt{6^2 + 18^2} = \sqrt{36 + 324} = \sqrt{360} = 6 \sqrt{10}.\)

Key Concepts

Vector Addition

Let's delve into vector addition! When you add two vectors, you combine their corresponding components.
This means taking the x-component of one vector and adding it to the x-component of the other vector. The same goes for the y-components.
To illustrate, consider vectors \( \mathbf{a} = \langle 1, 3 \rangle \) and \( \mathbf{b} = \langle -5, -15 \rangle \). When adding these vectors, you perform the following operations:

• For the x-component: \( 1 + (-5) = -4 \)
• For the y-component: \( 3 + (-15) = -12 \)

Therefore, \( \mathbf{a} + \mathbf{b} = \langle -4, -12 \rangle \).This resulting vector represents the combined influence or effect of the two original vectors.

Vector Magnitude

Understanding vector magnitude involves finding the length or size of a vector. This means measuring how long the vector is in the space it occupies.
The mathematical formula for the magnitude of a vector \( \mathbf{v} = \langle x, y \rangle \) is:\[\| \mathbf{v} \| = \sqrt{x^2 + y^2}\]For instance, the vector \( \mathbf{a} + \mathbf{b} = \langle -4, -12 \rangle \) has a magnitude calculated as follows:

• Find each component squared: \( (-4)^2 = 16 \), \( (-12)^2 = 144 \)
• Add these squares: \( 16 + 144 = 160 \)
• Take the square root: \( \sqrt{160} = 4 \sqrt{10} \)

This magnitude essentially tells you the distance of the vector from the origin in a coordinate plane.

Scalar Multiplication

Scalar multiplication is a way to scale a vector. You multiply every component of the vector by a constant, known as the scalar.
This operation stretches or shrinks the vector, making it longer or shorter while preserving its direction.
Take vector \( \mathbf{a} = \langle 1, 3 \rangle \) and a scalar 3. To find \( 3\mathbf{a} \), perform these multiplications:

• Multiply the x-component: \( 3 \times 1 = 3 \)
• Multiply the y-component: \( 3 \times 3 = 9 \)

Thus, \( 3\mathbf{a} = \langle 3, 9 \rangle \). This shows how the original vector \( \mathbf{a} \) is scaled by a factor of 3, making it three times longer, but still points in the same direction.

Vector Subtraction

Vector subtraction involves subtracting the components of one vector from another.
This operation is similar to addition but reverses the direction of what is being subtracted.
Consider vectors \( \mathbf{a} = \langle 1, 3 \rangle \) and \( \mathbf{b} = \langle -5, -15 \rangle \). To compute \( \mathbf{a} - \mathbf{b} \):

• Subtract the x-components: \( 1 - (-5) = 6 \)
• Subtract the y-components: \( 3 - (-15) = 18 \)

This results in \( \mathbf{a} - \mathbf{b} = \langle 6, 18 \rangle \). Subtracting vectors is like adding a vector with a negative version of the other, altering its direction but keeping the magnitude associated with their distance apart.