Problem 37 - Precalculus Mathematics for Calculus

Question

Find the direction angles of the given vector, rounded to the nearest degree. $$3 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k}$$

Step-by-Step Solution

Answer

The direction angles are approximately \(66^\circ, 55^\circ,\) and \(45^\circ\).

Step 1: Understand the Vector Components

The given vector is written in terms of unit vectors: it has components \(a = 3\), \(b = 4\), and \(c = 5\). We can represent this vector in the form \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k} \).

Step 2: Calculate the Magnitude of the Vector

The magnitude (or length) of the vector \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k} \) is calculated using the formula:\[|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2}\].

Step 3: Find the Cosine of Each Direction Angle

The direction angles \( \alpha, \beta, \gamma \) are the angles made with the \(x\)-, \(y\)-, and \(z\)-axes respectively. They are found using the cosines:\[ \cos \alpha = \frac{a}{|\mathbf{v}|} = \frac{3}{5\sqrt{2}}, \quad \cos \beta = \frac{b}{|\mathbf{v}|} = \frac{4}{5\sqrt{2}}, \quad \cos \gamma = \frac{c}{|\mathbf{v}|} = \frac{5}{5\sqrt{2}}\].

Step 4: Calculate the Direction Angles

The direction angles \( \alpha, \beta, \gamma \) are obtained by taking the inverse cosine (arccosine) of the respective cosine values:\[ \alpha = \cos^{-1}\left(\frac{3}{5\sqrt{2}}\right), \quad \beta = \cos^{-1}\left(\frac{4}{5\sqrt{2}}\right), \quad \gamma = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \].

Step 5: Convert and Round the Direction Angles

Using a calculator, evaluate the inverse cosines and convert the direction angles from radians to degrees, round to the nearest degree:\[ \alpha \approx 66^\circ, \quad \beta \approx 55^\circ, \quad \gamma \approx 45^\circ \].

Key Concepts

Vector Components

When dealing with vectors, understanding their components is essential. Vectors in three-dimensional space are often expressed in terms of unit vectors: \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). Each represents a direction along the x, y, and z axes, respectively.
The given vector \( 3 \mathbf{i} + 4 \mathbf{j} + 5 \mathbf{k} \) indicates that the vector has components:

• 3 in the x-direction (\( \mathbf{i} \))
• 4 in the y-direction (\( \mathbf{j} \))
• 5 in the z-direction (\( \mathbf{k} \))

These components are crucial for further calculations such as magnitude and direction angles of the vector.

Magnitude of a Vector

The magnitude of a vector gives the length or size of the vector from its initial point to its terminal point. For a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), the formula to calculate magnitude is:\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]Applying this formula to our vector, \( 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k} \), we have:

• \( a^2 = 3^2 = 9 \)
• \( b^2 = 4^2 = 16 \)
• \( c^2 = 5^2 = 25 \)

Adding these values gives:\[ 9 + 16 + 25 = 50 \]Thus, the magnitude is \( \sqrt{50} = 5\sqrt{2} \). Knowing the magnitude is essential for determining direction angles.

Cosine of an Angle

The cosine of an angle in a vector context helps define the angle's relationship with the axes. For a vector with components, the cosine of the direction angle (\( \alpha, \beta, \gamma \)) is calculated by comparing the component to the vector's magnitude:

• For angle \( \alpha \) with the x-axis: \( \cos \alpha = \frac{a}{|\mathbf{v}|} \)
• For angle \( \beta \) with the y-axis: \( \cos \beta = \frac{b}{|\mathbf{v}|} \)
• For angle \( \gamma \) with the z-axis: \( \cos \gamma = \frac{c}{|\mathbf{v}|} \)

For vector \( 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k} \), these values are\[ \cos \alpha = \frac{3}{5\sqrt{2}}, \quad \cos \beta = \frac{4}{5\sqrt{2}}, \quad \cos \gamma = \frac{5}{5\sqrt{2}} \].
This understanding allows us to discover the angle each vector component makes with the coordinate axes.

Inverse Cosine

Inverse cosine, also known as arccosine, is used to find the angle whose cosine is a specific value. In the context of vectors, if you know the cosine of a direction angle, using the inverse cosine function allows you to calculate the actual angle.

• \( \alpha = \cos^{-1}\left(\frac{3}{5\sqrt{2}}\right) \)
• \( \beta = \cos^{-1}\left(\frac{4}{5\sqrt{2}}\right) \)
• \( \gamma = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \)

Using a calculator, you can find these inverse cosine values and convert them into degrees. This gives the angles \( \alpha \approx 66^\circ \), \( \beta \approx 55^\circ \), and \( \gamma \approx 45^\circ \), which represent the angles between the vector and the coordinate axes.