Problem 31 - Advanced Engineering Mathematics

Question

Find a vector \(b\) that is parallel to the given vector and has the indicated magnitude. \(\mathbf{a}=3 \mathbf{i}+7 \mathbf{j},\|\mathbf{b}\|=2\)

Step-by-Step Solution

Answer

\(\mathbf{b} = \frac{6 \mathbf{i} + 14 \mathbf{j}}{\sqrt{58}}\)

Step 1: Find the Unit Vector of \( \mathbf{a} \)

First, we need to find the unit vector of \( \mathbf{a} \). A unit vector has a magnitude of 1 and points in the same direction as the original vector. The magnitude of \( \mathbf{a} \), denoted by \( \| \mathbf{a} \| \), is calculated as follows: \[ \| \mathbf{a} \| = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}. \] The unit vector \( \mathbf{u} \) for \( \mathbf{a} \) is given by: \[ \mathbf{u} = \frac{\mathbf{a}}{\| \mathbf{a} \|} = \frac{3 \mathbf{i} + 7 \mathbf{j}}{\sqrt{58}}. \]

Step 2: Scale the Unit Vector to Desired Magnitude

To find the vector \( \mathbf{b} \) that is parallel to \( \mathbf{a} \) and has the magnitude \( 2 \), we scale the unit vector \( \mathbf{u} \) by this magnitude. This is done by multiplying the unit vector by \( 2 \). Thus, \( \mathbf{b} \) is given by: \[ \mathbf{b} = 2 \cdot \mathbf{u} = 2 \cdot \frac{3 \mathbf{i} + 7 \mathbf{j}}{\sqrt{58}} = \frac{6 \mathbf{i} + 14 \mathbf{j}}{\sqrt{58}}. \] This vector \( \mathbf{b} \) will have the desired magnitude of \( 2 \) and is parallel to \( \mathbf{a} \).

Key Concepts

Unit Vector

A unit vector is a special kind of vector that has a magnitude of exactly 1 unit. It points in the same direction as the original vector but is scaled down to this specific length. The purpose of a unit vector is to provide a direction while standardizing its magnitude to one.
To find a unit vector for a given vector \( \mathbf{a} \), you start by calculating the magnitude or length of the vector. This is done using the formula for the magnitude of a vector, which sums the squares of its components, and then takes the square root of the result.
For example, if vector \( \mathbf{a} = 3 \mathbf{i} + 7 \mathbf{j} \), the magnitude is \( \| \mathbf{a} \| = \sqrt{3^2 + 7^2} = \sqrt{58} \).
The unit vector \( \mathbf{u} \) is then found by dividing the original vector by its magnitude:

• \( \mathbf{u} = \frac{\mathbf{a}}{\| \mathbf{a} \|} \)
• For our example, \( \mathbf{u} = \frac{3 \mathbf{i} + 7 \mathbf{j}}{\sqrt{58}} \)

This process ensures that the unit vector maintains the original direction of \( \mathbf{a} \) while having a length of 1.

Vector Magnitude

In vector mathematics, the magnitude of a vector is a measure of its length. It is also sometimes referred to as the 'norm' of the vector. The magnitude gives us an idea of how long the vector is and is a fundamental concept used across various applications such as physics, engineering, and computer graphics.
Calculating the magnitude involves using the Pythagorean theorem. For a two-dimensional vector \( \mathbf{a} = x \mathbf{i} + y \mathbf{j} \), the magnitude is represented as \( \| \mathbf{a} \| \) and calculated by:

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

For the vector \( \mathbf{a} = 3 \mathbf{i} + 7 \mathbf{j} \), its magnitude is \( \sqrt{3^2 + 7^2} = \sqrt{58} \).
This value, \( \sqrt{58} \), represents the distance from the origin to the point \( (3, 7) \) if the vector were to be represented in a coordinate system. It is this component that is crucial for normalization, which is the process of converting the vector into a unit vector.

Parallel Vectors

Parallel vectors are vectors that point in the same direction, although they can have different magnitudes. Think of them as arrows that follow the same path but may vary in length. These vectors essentially have the same or opposite direction. Parallelism in vectors is an important concept in determining vector relationships.
Vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if one is a scalar multiple of the other. This means there exists a number (scalar) \( k \) such that \( \mathbf{b} = k \mathbf{a} \).
For example, if \( \mathbf{a} = 3 \mathbf{i} + 7 \mathbf{j} \), and you need a vector \( \mathbf{b} \) with magnitude 2 that is parallel to \( \mathbf{a} \), you start by finding the unit vector of \( \mathbf{a} \). Then, \( \mathbf{b} \) is obtained by scaling this unit vector by the desired magnitude of 2:

• \( \mathbf{b} = 2 \cdot \mathbf{u} \)
• Where \( \mathbf{u} = \frac{3 \mathbf{i} + 7 \mathbf{j}}{\sqrt{58}} \)
• So, \( \mathbf{b} = \frac{6 \mathbf{i} + 14 \mathbf{j}}{\sqrt{58}} \)

This step preserves the direction provided by the unit vector while adjusting its magnitude, resulting in the desired parallel vector.