To prove that the points with the position vectors $\mathbf{A} = 2\hat{i} + 4\hat{j} + 3\hat{k}$, $\mathbf{B} = 4\hat{i} + \hat{j} + 9\hat{k}$, and $\mathbf{C} = 10\hat{i} - \hat{j} + 6\hat{k}$ form a right-angled triangle, we will first calculate the vectors representing the sides of the triangle, then check if the Pythagorean theorem holds.

Step 1: Find the vectors representing the sides of the triangle.

• The vector $\overrightarrow{AB}$ represents the side between points $A$ and $B$. $\overrightarrow{AB} = \mathbf{B} - \mathbf{A} = (4\hat{i} + \hat{j} + 9\hat{k}) - (2\hat{i} + 4\hat{j} + 3\hat{k}) = 2\hat{i} - 3\hat{j} + 6\hat{k}$

• The vector $\overrightarrow{BC}$ represents the side between points $B$ and $C$. $\overrightarrow{BC} = \mathbf{C} - \mathbf{B} = (10\hat{i} - \hat{j} + 6\hat{k}) - (4\hat{i} + \hat{j} + 9\hat{k}) = 6\hat{i} - 2\hat{j} - 3\hat{k}$

• The vector $\overrightarrow{CA}$ represents the side between points $C$ and $A$. $\overrightarrow{CA} = \mathbf{A} - \mathbf{C} = (2\hat{i} + 4\hat{j} + 3\hat{k}) - (10\hat{i} - \hat{j} + 6\hat{k}) = -8\hat{i} + 5\hat{j} - 3\hat{k}$

Step 2: Check if the triangle is right-angled using the dot product.

In a right-angled triangle, the vectors representing two sides meeting at a right angle will have a dot product of 0. We will compute the dot products of the pairs of vectors $\overrightarrow{AB}$, $\overrightarrow{BC}$, and $\overrightarrow{CA}$.

Dot product of $\overrightarrow{AB}$ and $\overrightarrow{BC}$:

$\overrightarrow{AB} \cdot \overrightarrow{BC} = (2\hat{i} - 3\hat{j} + 6\hat{k}) \cdot (6\hat{i} - 2\hat{j} - 3\hat{k})$ $= 2 \times 6 + (-3) \times (-2) + 6 \times (-3) = 12 + 6 - 18 = 0$ Since the dot product is 0, the vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are perpendicular, meaning the angle between them is 90°.

Step 3: Conclusion.

Since the dot product of $\overrightarrow{AB}$ and $\overrightarrow{BC}$ is 0, the triangle formed by the points $A$, $B$, and $C$ is a right-angled triangle with the right angle at point $B$.

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Would you like further details or have any questions about this solution?

Further Questions:

1. How do you calculate the length of the sides of a triangle given the position vectors of the vertices?
2. What are the properties of a right-angled triangle in vector geometry?
3. How can the cross product be used to check for perpendicular vectors in three-dimensional space?
4. How do you apply the law of cosines in vector form?
5. How do you find the area of a right-angled triangle using vectors?

Tip: The dot product is an essential tool for checking orthogonality (perpendicularity) of vectors, which is useful in many geometry problems.