To find an orthogonal vector to the given vector $\mathbf{v} = (2, -3, 2, 1)$, we need to find a vector $\mathbf{w} = (w_1, w_2, w_3, w_4)$ such that the dot product of $\mathbf{v}$ and $\mathbf{w}$ equals zero:

$\mathbf{v} \cdot \mathbf{w} = 2w_1 + (-3)w_2 + 2w_3 + 1w_4 = 0$

Let's go through the steps to find such a vector.

Step 1: Write the equation for orthogonality

The dot product of $\mathbf{v}$ and $\mathbf{w}$ is: $2w_1 - 3w_2 + 2w_3 + w_4 = 0$ We now need to find values for $w_1, w_2, w_3,$ and $w_4$ that satisfy this equation.

Step 2: Choose values for some of the components

We can choose arbitrary values for some components of $\mathbf{w}$ because there are infinitely many orthogonal vectors in 4-dimensional space.

Let's choose $w_1 = 1$ and $w_2 = 0$. This will simplify the equation.

Step 3: Solve for $w_3$ and $w_4$

Substituting $w_1 = 1$ and $w_2 = 0$ into the equation $2w_1 - 3w_2 + 2w_3 + w_4 = 0$, we get: $2(1) - 3(0) + 2w_3 + w_4 = 0$ This simplifies to: $2 + 2w_3 + w_4 = 0$ $2w_3 + w_4 = -2$ Now, we can choose one of the variables and solve for the other. Let's choose $w_3 = 1$.

Substituting $w_3 = 1$ into the equation $2w_3 + w_4 = -2$: $2(1) + w_4 = -2$ $2 + w_4 = -2$ $w_4 = -4$

Step 4: Write the orthogonal vector

We now have the components of $\mathbf{w}$: $w_1 = 1, \quad w_2 = 0, \quad w_3 = 1, \quad w_4 = -4$ So, one possible orthogonal vector to $\mathbf{v} = (2, -3, 2, 1)$ is: $\mathbf{w} = (1, 0, 1, -4)$

Verification

To verify, we can compute the dot product: $\mathbf{v} \cdot \mathbf{w} = 2(1) + (-3)(0) + 2(1) + 1(-4) = 2 + 0 + 2 - 4 = 0$ Since the dot product is zero, the vector $\mathbf{w} = (1, 0, 1, -4)$ is indeed orthogonal to $\mathbf{v}$.

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Would you like more details or have any further questions?

Here are 5 related questions to further your understanding:

1. What is the geometric meaning of orthogonal vectors in higher dimensions?
2. Can you find another orthogonal vector to $\mathbf{v} = (2, -3, 2, 1)$?
3. How would you find an orthogonal vector if you were in a 3-dimensional space?
4. What is the significance of dot products in physics or engineering?
5. Can two vectors be orthogonal if their magnitudes are different?

Tip: In $\mathbb{R}^n$, the set of vectors orthogonal to a given vector forms a hyperplane.