if i have a got a question like:
Show that the points with the position vectors 2i+3j+(√3)k, (√10)i-j+(√5)k and -3i+(√3)j+2k are vectors of equilateral triangle. How will u solve it

To solve this problem, you need to show that the given points form an equilateral triangle. This involves proving that the distances (magnitudes of the vectors) between each pair of points are equal. Here's a step-by-step method to solve this:

### Given Vectors
- **Vector A**: \( \mathbf{A} = 2\mathbf{i} + 3\mathbf{j} + \sqrt{3}\mathbf{k} \)
- **Vector B**: \( \mathbf{B} = \sqrt{10}\mathbf{i} - \mathbf{j} + \sqrt{5}\mathbf{k} \)
- **Vector C**: \( \mathbf{C} = -3\mathbf{i} + \sqrt{3}\mathbf{j} + 2\mathbf{k} \)

### Step 1: Calculate the Vector Differences
Find the vectors representing the sides of the triangle by subtracting the vectors:

1. **Vector \( \mathbf{AB} \)**:
   \[
   \mathbf{AB} = \mathbf{B} - \mathbf{A} = (\sqrt{10} - 2)\mathbf{i} + (-1 - 3)\mathbf{j} + (\sqrt{5} - \sqrt{3})\mathbf{k}
   \]

2. **Vector \( \mathbf{BC} \)**:
   \[
   \mathbf{BC} = \mathbf{C} - \mathbf{B} = (-3 - \sqrt{10})\mathbf{i} + (\sqrt{3} + 1)\mathbf{j} + (2 - \sqrt{5})\mathbf{k}
   \]

3. **Vector \( \mathbf{CA} \)**:
   \[
   \mathbf{CA} = \mathbf{A} - \mathbf{C} = (2 + 3)\mathbf{i} + (3 - \sqrt{3})\mathbf{j} + (\sqrt{3} - 2)\mathbf{k}
   \]

### Step 2: Calculate the Magnitudes of the Vectors
Find the magnitude (length) of each vector:

1. **Magnitude of \( \mathbf{AB} \)**:
   \[
   |\mathbf{AB}| = \sqrt{(\sqrt{10} - 2)^2 + (-4)^2 + (\sqrt{5} - \sqrt{3})^2}
   \]

2. **Magnitude of \( \mathbf{BC} \)**:
   \[
   |\mathbf{BC}| = \sqrt{(-3 - \sqrt{10})^2 + (\sqrt{3} + 1)^2 + (2 - \sqrt{5})^2}
   \]

3. **Magnitude of \( \mathbf{CA} \)**:
   \[
   |\mathbf{CA}| = \sqrt{(5)^2 + (3 - \sqrt{3})^2 + (\sqrt{3} - 2)^2}
   \]

### Step 3: Show that the Magnitudes are Equal
For the points to form an equilateral triangle, the magnitudes of \( \mathbf{AB} \), \( \mathbf{BC} \), and \( \mathbf{CA} \) must be equal:
\[
|\mathbf{AB}| = |\mathbf{BC}| = |\mathbf{CA}|
\]

### Step 4: Simplify and Compare
Simplify the expressions for the magnitudes and check if they are indeed equal. If they are, then the vectors form an equilateral triangle.

### Summary
If the magnitudes are equal, the given points form an equilateral triangle. If not, they do not form an equilateral triangle.

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Do you want to see the actual calculation for these magnitudes? Or any other questions related to this?

### Follow-Up Questions:
1. How do you calculate the magnitude of a vector?
2. What is the geometric interpretation of vector subtraction?
3. Can you explain why equal magnitudes imply an equilateral triangle?
4. What are the conditions for three points to form an equilateral triangle?
5. How would the approach differ if one of the vectors was a zero vector?
6. What if the points form an isosceles triangle instead? How would you check that?
7. Can you provide a different set of vectors that form an equilateral triangle?
8. How does the vector cross product relate to checking if points form an equilateral triangle?

### Tip:
When dealing with vector problems, visualizing the points in a 3D coordinate system can help you understand the relationships between the vectors better.

Learn how to prove that given points form an equilateral triangle using vector calculations. This problem involves calculating vector differences and magnitudes to verify equal distances between points in a 3D space.

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