The direction cosines and direction angles of a vector $\mathbf{v} = \langle 2, 1, 2 \rangle$ can be determined by first finding the magnitude of the vector, and then using it to compute the cosines of the angles between the vector and the coordinate axes. These angles are called the direction angles.

Step 1: Find the magnitude of the vector

The magnitude $|\mathbf{v}|$ of the vector $\mathbf{v} = \langle 2, 1, 2 \rangle$ is given by:

$|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$

Step 2: Find the direction cosines

The direction cosines are given by:

$\cos \alpha = \frac{x}{|\mathbf{v}|}, \quad \cos \beta = \frac{y}{|\mathbf{v}|}, \quad \cos \gamma = \frac{z}{|\mathbf{v}|}$

where $\alpha$, $\beta$, and $\gamma$ are the direction angles with respect to the $x$-, $y$-, and $z$-axes, respectively.

For the vector $\mathbf{v} = \langle 2, 1, 2 \rangle$ and its magnitude $|\mathbf{v}| = 3$:

$\cos \alpha = \frac{2}{3}, \quad \cos \beta = \frac{1}{3}, \quad \cos \gamma = \frac{2}{3}$

Step 3: Find the direction angles

To find the direction angles, take the inverse cosine (arccos) of the direction cosines:

$\alpha = \cos^{-1} \left( \frac{2}{3} \right), \quad \beta = \cos^{-1} \left( \frac{1}{3} \right), \quad \gamma = \cos^{-1} \left( \frac{2}{3} \right)$

Now, calculate these values to the nearest degree:

$\alpha = \cos^{-1} \left( \frac{2}{3} \right) \approx 48^\circ$ $\beta = \cos^{-1} \left( \frac{1}{3} \right) \approx 70^\circ$ $\gamma = \cos^{-1} \left( \frac{2}{3} \right) \approx 48^\circ$

Final Answer:

• Direction cosines: $\cos \alpha = \frac{2}{3}$, $\cos \beta = \frac{1}{3}$, $\cos \gamma = \frac{2}{3}$
• Direction angles: $\alpha = 48^\circ$, $\beta = 70^\circ$, $\gamma = 48^\circ$

Would you like further details or have any questions?

Here are some related questions:

1. What are direction cosines for vectors in 2D space?
2. How do you interpret direction angles geometrically?
3. What is the difference between direction angles and direction cosines?
4. How are direction cosines used in physics, especially in vectors in 3D space?
5. What would happen to direction angles if the vector were normalized?

Tip: The direction angles always add up to less than $180^\circ$ because they are the angles between the vector and each axis, not the angles between the axes themselves.