for any θ, the vectors U=cosθI+sinθJ, V=-sinθI+cosθJ,

form an orthonormal basis for the plane; that is, they are orthogonal vectors of length 1.

Let θ= 2pi/5, and X=10I+9J. Then we can write X=uU+vV 

with u= ?, v=?

Given the vectors \( U = \cos(\theta)I + \sin(\theta)J \) and \( V = -\sin(\theta)I + \cos(\theta)J \), and the vector \( X = 10I + 9J \), we aim to express \( X \) in terms of the orthonormal basis vectors \( U \) and \( V \).

We can write:

\[
X = uU + vV
\]

This means we need to find the coefficients \( u \) and \( v \). Since \( U \) and \( V \) are orthonormal, we can use the dot product to find \( u \) and \( v \):

\[
u = X \cdot U
\]
\[
v = X \cdot V
\]

Now, let's compute these values step by step.

### Step 1: Calculate \( U \) and \( V \) when \( \theta = \frac{2\pi}{5} \)

- \( \cos\left(\frac{2\pi}{5}\right) = \cos(72^\circ) \)
- \( \sin\left(\frac{2\pi}{5}\right) = \sin(72^\circ) \)

Using these values:

\[
U = \cos\left(\frac{2\pi}{5}\right) I + \sin\left(\frac{2\pi}{5}\right) J
\]
\[
V = -\sin\left(\frac{2\pi}{5}\right) I + \cos\left(\frac{2\pi}{5}\right) J
\]

### Step 2: Compute the dot products

Let's calculate \( u = X \cdot U \) and \( v = X \cdot V \).

I'll now compute these values.The values for \( u \) and \( v \) are:

\[
u \approx 11.65
\]
\[
v \approx -6.73
\]

Thus, the vector \( X = 10I + 9J \) can be written as:

\[
X = 11.65U - 6.73V
\]

Would you like further details or clarifications?

Here are five related questions to explore:
1. What is the geometric interpretation of orthonormal vectors?
2. How can we generalize this result for any angle \( \theta \)?
3. What happens to \( U \) and \( V \) as \( \theta \) approaches \( 0 \)?
4. Can you find \( u \) and \( v \) for different values of \( \theta \), such as \( \frac{\pi}{4} \)?
5. How does changing the vector \( X \) affect the values of \( u \) and \( v \)?

**Tip**: Orthonormal vectors simplify vector decompositions, as dot products with these vectors directly give you the required coefficients.

For any θ, the vectors U = cosθI + sinθJ, V = -sinθI + cosθJ, form an orthonormal basis for the plane; that is, they are orthogonal vectors of length 1. Let θ = 2π/5, and X = 10I + 9J. Then we can write X = uU + vV with u = ?, v = ?

Learn how to decompose the vector X = 10I + 9J using the orthonormal basis U and V with θ = 2π/5. This problem covers vector decomposition, trigonometric calculations, and dot products. Suitable for university-level students studying linear algebra.

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