Problem 49 - Advanced Engineering Mathematics

Question

Find a unit vector in the opposite direction of \(\mathbf{a}=\langle 10,-5,10\rangle\).

Step-by-Step Solution

Answer

The unit vector in the opposite direction is \(-\langle \frac{2}{3}, \frac{1}{3}, \frac{2}{3} \rangle\).

Step 1: Calculate the Magnitude of Vector

First, we need to find the magnitude (or length) of the vector \(\mathbf{a} = \langle 10, -5, 10 \rangle\). The magnitude is calculated using the formula:\[||\mathbf{a}|| = \sqrt{10^2 + (-5)^2 + 10^2}\]Calculating this, we get:\[||\mathbf{a}|| = \sqrt{100 + 25 + 100} = \sqrt{225} = 15\]

Step 2: Calculate the Unit Vector

A unit vector in the same direction as \(\mathbf{a}\) is found by dividing each component of \(\mathbf{a}\) by its magnitude. The unit vector \(\mathbf{u}\) is:\[\mathbf{u} = \left\langle \frac{10}{15}, \frac{-5}{15}, \frac{10}{15} \right\rangle = \langle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \rangle\]

Step 3: Find the Opposite Direction Vector

To find a unit vector in the opposite direction, we multiply each component of \(\mathbf{u}\) by \(-1\). This gives:\[-\mathbf{u} = \left\langle -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \right\rangle\]

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of 1. It points in a particular direction without being concerned with the vector's length. We often use unit vectors to indicate direction. In mathematical terms, we find a unit vector by dividing a given vector by its magnitude. This process scales the vector to a magnitude of one, effectively preserving the direction but not its original length.
For example, given a vector \( \mathbf{a} = \langle 10, -5, 10 \rangle \), we calculate the unit vector by dividing each of its components by the magniture of \( \mathbf{a} \).

• Divide each component of the vector by its magnitude.
• This process maintains the direction while setting the length to 1.

Unit vectors are helpful when analyzing vector directions independently of their magnitudes, such as when creating direction vectors in physics or computer graphics.

Magnitude Calculation

The magnitude of a vector is the length or norm of the vector. To find the magnitude of a vector \( \mathbf{a} = \langle x, y, z \rangle \), we use the formula:

\[ ||\mathbf{a}|| = \sqrt{x^2 + y^2 + z^2} \]

This formula derives from the Pythagorean theorem, extended into three-dimensional space.
Using our example \( \mathbf{a} = \langle 10, -5, 10 \rangle \), we find its magnitude as follows:

• First, square each component: \( 10^2 = 100 \), \( (-5)^2 = 25 \), \( 10^2 = 100 \).
• Sum these squares: \( 100 + 25 + 100 = 225 \).
• Finally, take the square root of this sum to obtain the magnitude: \( \sqrt{225} = 15 \).

The magnitude gives us insight into the size or length of the vector, which is key when determining a unit vector.

Vector Direction

Vector direction represents where the vector points. Direction is an essential part of vector analysis, denoting the orientation in space. While the magnitude provides the vector's length, the direction denotes its trajectory.

To find the direction of the opposite of a given vector, we simply change the signs of each component of its unit vector. This mirrors the direction.
Using the unit vector \( \mathbf{u} = \langle \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \rangle \) from our previous example:

• Multiply each component by \(-1\): \( -\frac{2}{3} \), \( \frac{1}{3} \), \( -\frac{2}{3} \).

This provides the unit vector in the opposite direction: \( -\mathbf{u} = \langle -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \rangle \).

Understanding vector direction is vital in fields like physics and engineering, where the orientation of vectors significantly impacts the systems being studied.