To normalize the given orthogonal set in $\mathbb{R}^3$, we need to calculate the magnitudes of each vector and then divide each vector by its magnitude to find the normalization constants $a_1$, $a_2$, and $a_3$.

The vectors in question are:

1. $\mathbf{v_1} = \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$
2. $\mathbf{v_2} = \begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix}$
3. $\mathbf{v_3} = \begin{pmatrix} 4 \\ 7 \\ 1 \end{pmatrix}$

Steps to Normalize Each Vector

1. Find the magnitude of each vector $\mathbf{v_i}$: $\| \mathbf{v_i} \| = \sqrt{x^2 + y^2 + z^2}$
2. Calculate normalization constants $a_1$, $a_2$, and $a_3$ by setting $a_i = \frac{1}{\| \mathbf{v_i} \|}$.

Let's go through each calculation.

1. Calculating $a_1$ for $\mathbf{v_1}$

$\| \mathbf{v_1} \| = \sqrt{1^2 + (-1)^2 + 3^2} = \sqrt{1 + 1 + 9} = \sqrt{11}$ $a_1 = \frac{1}{\sqrt{11}}$

2. Calculating $a_2$ for $\mathbf{v_2}$

$\| \mathbf{v_2} \| = \sqrt{(-2)^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$ $a_2 = \frac{1}{\sqrt{6}}$

3. Calculating $a_3$ for $\mathbf{v_3}$

$\| \mathbf{v_3} \| = \sqrt{4^2 + 7^2 + 1^2} = \sqrt{16 + 49 + 1} = \sqrt{66}$ $a_3 = \frac{1}{\sqrt{66}}$

Final Answer

So, the values for the normalization constants are:

• $a_1 = \frac{1}{\sqrt{11}}$
• $a_2 = \frac{1}{\sqrt{6}}$
• $a_3 = \frac{1}{\sqrt{66}}$

Would you like more details on any specific step?

Here are five related questions to explore:

1. How would you normalize a set of vectors in $\mathbb{R}^2$ instead?
2. What is the significance of orthogonal vectors in vector spaces?
3. Can you verify if two vectors are orthogonal by checking their dot product?
4. How do you find an orthonormal basis for a subspace?
5. What is the Gram-Schmidt process, and how is it used in orthogonalizing vectors?

Tip: Normalizing a vector turns it into a unit vector, which is helpful in various applications like defining directions in physics and engineering.