Let's work through the problems shown in the image step by step.

Problem 4:

You are given:

• $x = (2, 1 + i, i)$
• $y = (2 - i, 1, 1 + 2i)$

The goal is to find the inner product $\langle x, y \rangle$.

For complex vectors, the inner product $\langle x, y \rangle$ is defined as:

$\langle x, y \rangle = \sum_{k=1}^{n} x_k \cdot \overline{y_k}$ where $\overline{y_k}$ is the complex conjugate of the corresponding element in $y$.

Now compute step-by-step:

• $x_1 = 2$ and $y_1 = 2 - i$, so: $x_1 \cdot \overline{y_1} = 2 \cdot (2 + i) = 4 + 2i$
• $x_2 = 1 + i$ and $y_2 = 1$, so: $x_2 \cdot \overline{y_2} = (1 + i) \cdot 1 = 1 + i$
• $x_3 = i$ and $y_3 = 1 + 2i$, so: $x_3 \cdot \overline{y_3} = i \cdot (1 - 2i) = i - 2i^2 = i + 2 = 2 + i$

Now sum them up: $\langle x, y \rangle = (4 + 2i) + (1 + i) + (2 + i) = 7 + 4i$

Thus, $\langle x, y \rangle = 7 + 4i$.

Problem 5:

You are given:

• $x = (1, 2, -3)$
• $y = (-1, 4, -3)$

Check if $x$ and $y$ are orthogonal by verifying whether $\langle x, y \rangle = 0$.

The inner product for real vectors is simply: $\langle x, y \rangle = x_1 y_1 + x_2 y_2 + x_3 y_3$ Now compute: $\langle x, y \rangle = (1 \cdot -1) + (2 \cdot 4) + (-3 \cdot -3) = -1 + 8 + 9 = 16$

Since $\langle x, y \rangle = 16 \neq 0$, the vectors $x$ and $y$ are not orthogonal.

Problem 6:

You are given:

• $u = (3, 2, k, -5)$
• $v = (1, k, 7, 3)$

Find the value of $k$ such that $u$ and $v$ are orthogonal, i.e., $\langle u, v \rangle = 0$.

The inner product is: $\langle u, v \rangle = 3 \cdot 1 + 2 \cdot k + k \cdot 7 + (-5) \cdot 3$ Simplifying: $\langle u, v \rangle = 3 + 2k + 7k - 15 = 2k + 7k - 12 = 9k - 12$ Set $\langle u, v \rangle = 0$ to find $k$: $9k - 12 = 0 \implies 9k = 12 \implies k = \frac{12}{9} = \frac{4}{3}$

Thus, $k = \frac{4}{3}$.

Would you like details or have any questions on any of the steps?

Here are 5 related questions:

1. How does the concept of orthogonality extend to higher-dimensional spaces?
2. What is the difference between inner product spaces with real and complex vectors?
3. Can two vectors be orthogonal if their components are complex numbers?
4. What are the geometrical interpretations of the inner product?
5. How can you apply the Cauchy-Schwarz inequality to check vector relationships?

Tip: When dealing with complex vectors, always remember to take the conjugate of the second vector in the inner product calculation.