Problem 29 - Advanced Engineering Mathematics

Question

\( \mathbf{a}=\langle 2,8\rangle\) and \(\mathbf{b}=\langle 3,4\rangle .\) Find a unit vector in the same direction as the given vector. \(\mathbf{a}+\mathbf{b}\)

Step-by-Step Solution

Answer

The unit vector is \( \left\langle \frac{5}{13}, \frac{12}{13} \right\rangle \)."

Step 1: Find the Resultant Vector

First, we need to calculate the vector \( \mathbf{c} \) which is the sum of the two given vectors \( \mathbf{a} \) and \( \mathbf{b} \). We do this by adding their corresponding components: \( \mathbf{c} = \langle 2, 8 \rangle + \langle 3, 4 \rangle = \langle 2+3, 8+4 \rangle = \langle 5, 12 \rangle \).

Step 2: Calculate the Magnitude of the Resultant Vector

To find a unit vector in the direction of \( \mathbf{c} = \langle 5, 12 \rangle \), we need the magnitude of \( \mathbf{c} \). This is given by the formula: \( \| \mathbf{c} \| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).

Step 3: Find the Unit Vector

A unit vector in the direction of \( \mathbf{c} \) is obtained by dividing each component of \( \mathbf{c} \) by its magnitude. Thus, the unit vector is \( \mathbf{u} = \left\langle \frac{5}{13}, \frac{12}{13} \right\rangle \).

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude (or length) of one. It maintains the same direction as the original vector but is scaled down to have a length of precisely one unit. To find a unit vector, you divide each component of the original vector by the vector's magnitude. This process ensures that the resulting unit vector has a magnitude of one while pointing in the same direction. For example, if you have a vector \( \mathbf{c} = \langle 5, 12 \rangle \), its unit vector is obtained by dividing each of its components by its magnitude, which we calculated as 13. Hence, the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{c} \) is:\[ \mathbf{u} = \left\langle \frac{5}{13}, \frac{12}{13} \right\rangle \]Remember:

• A unit vector always has a magnitude of one.
• It simply changes the original vector's length while keeping its direction.

Resultant Vector

A resultant vector is the vector sum of two or more vectors. It represents the combined effect of multiple vectors applied in sequence. Calculating a resultant vector involves adding the corresponding components of each vector being combined.Let's look at an example with the vectors \( \mathbf{a} = \langle 2, 8 \rangle \) and \( \mathbf{b} = \langle 3, 4 \rangle \):- To find the resultant vector \( \mathbf{c} \), just add their corresponding components. - So, \( \mathbf{c} = \langle 2 + 3, 8 + 4 \rangle = \langle 5, 12 \rangle \).The resultant vector \( \mathbf{c} \) shows the overall direction and magnitude of \( \mathbf{a} \) and \( \mathbf{b} \) combined. Consider it like adding forces together to see the total impact.

• The formula involves straightforward addition of each component.
• Analyzing resultant vectors is essential in physics and engineering for determining net effects.

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size. It is calculated using the Pythagorean theorem, applied in a two-dimensional space as:\[ \| \mathbf{v} \| = \sqrt{x^2 + y^2} \]Where \( x \) and \( y \) are the components of the vector \( \mathbf{v} \).For example, for a vector \( \mathbf{c} = \langle 5, 12 \rangle \), you can find its magnitude by:- Plugging into the formula to get \( \| \mathbf{c} \| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).The magnitude gives you the distance from the origin to the point represented by the vector in the coordinate plane. Key points:

• It is always non-negative.
• Reflects the true extent of the vector, different from direction.

Understanding the magnitude is crucial for tasks like normalizing vectors or determining lengths in geometry and science.