Problem 28 - Advanced Engineering Mathematics

Question

Find the direction cosines and direction angles of the given vector. $$ \mathbf{a}=\langle 5,7,2\rangle $$

Step-by-Step Solution

Answer

Direction cosines are \( \left(\frac{5}{\sqrt{78}}, \frac{7}{\sqrt{78}}, \frac{2}{\sqrt{78}}\right) \). Direction angles are \( \alpha = \cos^{-1}\left(\frac{5}{\sqrt{78}}\right), \beta = \cos^{-1}\left(\frac{7}{\sqrt{78}}\right), \gamma = \cos^{-1}\left(\frac{2}{\sqrt{78}}\right) \).

Step 1: Find the Magnitude of the Vector

The magnitude of vector \( \mathbf{a} = \langle 5, 7, 2 \rangle \) is found using the formula: \[ |\mathbf{a}| = \sqrt{x^2 + y^2 + z^2} \]Substitute the components of \( \mathbf{a} \): \[ |\mathbf{a}| = \sqrt{5^2 + 7^2 + 2^2} = \sqrt{25 + 49 + 4} = \sqrt{78} \]

Step 2: Calculate Direction Cosines

The direction cosines \( (l, m, n) \) can be found using:\[ l = \frac{x}{|\mathbf{a}|}, \quad m = \frac{y}{|\mathbf{a}|}, \quad n = \frac{z}{|\mathbf{a}|} \]Substitute the vector components and magnitude:\[ l = \frac{5}{\sqrt{78}}, \quad m = \frac{7}{\sqrt{78}}, \quad n = \frac{2}{\sqrt{78}} \]

Step 3: Calculate Direction Angles

The direction angles \( \alpha, \beta, \gamma \) with the x, y, and z axes are found using the inverse cosines of the direction cosines:\[ \alpha = \cos^{-1}(l), \beta = \cos^{-1}(m), \gamma = \cos^{-1}(n) \]Substitute the calculated direction cosines:\[ \alpha = \cos^{-1}\left(\frac{5}{\sqrt{78}}\right), \beta = \cos^{-1}\left(\frac{7}{\sqrt{78}}\right), \gamma = \cos^{-1}\left(\frac{2}{\sqrt{78}}\right) \]

Key Concepts

Magnitude of a Vector

Understanding the magnitude of a vector is crucial when working with vectors in 3D space. The magnitude represents the length or size of the vector. For a vector \( \mathbf{a} = \langle x, y, z \rangle \), the magnitude is found using the formula: \[|mathbf{a}|= \sqrt{x^2 + y^2 + z^2}\] This formula stems from the Pythagorean theorem and extends it into three dimensions.
To find the magnitude, simply square each component, add them together, and then take the square root of the result.

• Squaring each component ensures that the magnitude is always a positive value, as it represents a physical length.
• For \( \mathbf{a} = \langle 5, 7, 2 \rangle \): - Square the components: \(5^2 = 25\), \(7^2 = 49\), \(2^2 = 4\). - Add them together: \(25 + 49 + 4 = 78\). - Take the square root: \(\sqrt{78}\).

So, the magnitude \(|\mathbf{a}| = \sqrt{78}\). This measurement allows you to quantitatively describe how large a vector is, regardless of its direction.

Direction Angles

Direction angles are angles that a vector forms with each of the coordinate axes. Specifically, these are typically the angles with the x-axis, y-axis, and z-axis. We denote them as \(\alpha, \beta,\) and \(\gamma\) respectively.
To find these direction angles, we utilize the concept of direction cosines, which relate to the vector's orientation.

• Calculate the direction cosines \((l, m, n)\) as ratios of the vector components to the magnitude. - \(l = \frac{x}{|\mathbf{a}|}\) - \(m = \frac{y}{|\mathbf{a}|}\) - \(n = \frac{z}{|\mathbf{a}|}\)
• The direction angles are then calculated using: - \(\alpha = \cos^{-1}(l)\) - \(\beta = \cos^{-1}(m)\) - \(\gamma = \cos^{-1}(n)\)
• For \(\mathbf{a} = \langle 5, 7, 2 \rangle\), given \(|\mathbf{a}| = \sqrt{78}\): - \(l = \frac{5}{\sqrt{78}}\) - \(m = \frac{7}{\sqrt{78}}\) - \(n = \frac{2}{\sqrt{78}}\)
• Use inverse cosine to find \(\alpha, \beta,\) and \(\gamma\): - \(\alpha = \cos^{-1}\left(\frac{5}{\sqrt{78}}\right)\) - \(\beta = \cos^{-1}\left(\frac{7}{\sqrt{78}}\right)\) - \(\gamma = \cos^{-1}\left(\frac{2}{\sqrt{78}}\right)\)

Direction angles give us a way to describe how a vector is oriented in relation to each axis.

Vector Components

Vector components are the building blocks of vectors in 3D space. Each vector in three dimensions can be expressed using its respective components along the x, y, and z axes.
The vector \( \mathbf{a} = \langle 5, 7, 2 \rangle \) tells us precisely how much we move in the direction of each axis.

• The x-component is \(5\), indicating movement along the x-axis.
• The y-component is \(7\), representing movement along the y-axis.
• The z-component is \(2\), showing movement along the z-axis.

To better understand vector components, visualize them as small right angle movements necessary to reach a point in 3D space. These components are often used in physics and engineering to decompose forces and velocity vectors.
Components are helpful for calculating other vector properties, like magnitude and direction angles, as seen in the solutions.
By understanding each component's value, you can accurately predict the direction and strength of the vector's overall effect.