give me an orthogonal vector for the vector (2,-3,2,1) and take me through the steps



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 \( \mathbf{v} \).

---

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.

give me an orthogonal vector for the vector (2,-3,2,1) and take me through the steps

Discover how to find an orthogonal vector to (2, -3, 2, 1) through a step-by-step solution. This problem involves key linear algebra concepts such as orthogonality, the dot product, and vector spaces. Perfect for high school students in Grades 10-12.

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