Problem 34 - Advanced Engineering Mathematics

Question

Given that \(\mathbf{a}=\langle 1,1\rangle\) and \(\mathbf{b}=\langle-1,0\rangle\), find a vector in the same direction as \(\mathbf{a}+\mathbf{b}\) but five times as long.

Step-by-Step Solution

Answer

The vector is \( \langle 0, 5 \rangle \).

Step 1: Find the Sum of Vectors

First, we need to add the vectors \( \mathbf{a} = \langle 1, 1 \rangle \) and \( \mathbf{b} = \langle -1, 0 \rangle \). The sum of these vectors is found by adding their corresponding components:\[ \mathbf{a} + \mathbf{b} = \langle 1 + (-1), 1 + 0 \rangle = \langle 0, 1 \rangle \]

Step 2: Determine the Magnitude of the Sum Vector

Next, let's determine the magnitude of the vector \( \mathbf{a} + \mathbf{b} = \langle 0, 1 \rangle \). The magnitude is calculated using the formula for the magnitude of a vector:\[ ||\mathbf{a} + \mathbf{b}|| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \]

Step 3: Normalize the Vector

To find a vector in the same direction but with a different magnitude, we start by finding the unit vector in the direction of \( \mathbf{a} + \mathbf{b} \). The unit vector is given by dividing each component by the magnitude:\[ \frac{\langle 0, 1 \rangle}{1} = \langle 0, 1 \rangle \]

Step 4: Scale the Vector by Five

Finally, we scale the unit vector \( \langle 0, 1 \rangle \) by five to get a vector that is five times as long but in the same direction:\[ 5 \times \langle 0, 1 \rangle = \langle 0, 5 \rangle \]

Key Concepts

Vector Addition

When dealing with vectors, one common operation is vector addition, where you combine two or more vectors to get a resultant vector. Imagine vectors as arrows in a graph. To add them, you arrange them so that the tail of one vector connects to the head of another.
For instance, if you have vector \( \mathbf{a} = \langle 1, 1 \rangle \) and vector \( \mathbf{b} = \langle -1, 0 \rangle \), you add their corresponding components:

• Horizontal component: \( 1 + (-1) = 0 \)
• Vertical component: \( 1 + 0 = 1 \)

The resulting vector is \( \langle 0, 1 \rangle \). This new vector visually represents the combined effect of the original vectors.

Vector Magnitude

The magnitude of a vector is like its length. Just as you measure the length of a line segment, you find the magnitude of a vector.
It tells you how long the vector is, without considering its direction. For a 2D vector, \( \mathbf{v} = \langle x, y \rangle \), the magnitude \( ||\mathbf{v}|| \) is calculated using the formula:

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

Using the vector \( \langle 0, 1 \rangle \) from our example, the magnitude is:

• \( ||\langle 0, 1 \rangle|| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \)

Thus, this vector has a length of 1.

Unit Vector

A unit vector has a magnitude of 1 and indicates direction. It is a handy way to work with direction separate from length.
By normalizing a vector, you transform it into a unit vector. This is done by dividing each component of the vector by its magnitude.
For the vector \( \langle 0, 1 \rangle \):

• Since the magnitude is 1, the unit vector is simply \( \frac{\langle 0, 1 \rangle}{1} = \langle 0, 1 \rangle \)

Unit vectors are crucial in various calculations, especially when you want to scale the vector later while keeping the same direction.

Vector Scaling

Vector scaling is the process of changing the length of a vector while keeping its direction unchanged. It's like stretching or shrinking the vector.
If you have a unit vector, you can scale it to any desired length by multiplying it by a scalar (a number).
From the example, the unit vector \( \langle 0, 1 \rangle \) can be scaled by 5 to get:

• \( 5 \times \langle 0, 1 \rangle = \langle 0, 5 \rangle \)

This scaling process leads to a vector that maintains its original direction but becomes five times as long.