To determine whether the vectors a = (4, 0, -1), b = (1, 3, 4), and c = (-5, -3, -3) form the sides of a triangle, we need to do the following:

1. Check if they form a triangle: To confirm if the vectors form a triangle, we need to check if the sum of any two vectors equals the third vector (which would imply they are collinear, thus not forming a triangle). For this, we calculate the vectors representing the sides of the triangle as follows:

   Let: $\text{Side } \mathbf{AB} = \mathbf{b} - \mathbf{a} = (1 - 4, 3 - 0, 4 - (-1)) = (-3, 3, 5)$ $\text{Side } \mathbf{BC} = \mathbf{c} - \mathbf{b} = (-5 - 1, -3 - 3, -3 - 4) = (-6, -6, -7)$ $\text{Side } \mathbf{CA} = \mathbf{a} - \mathbf{c} = (4 - (-5), 0 - (-3), -1 - (-3)) = (9, 3, 2)$

   We now check if any of these vectors are collinear:

   • For vectors $\mathbf{AB}$ and $\mathbf{BC}$, check if $\mathbf{AB} = k \cdot \mathbf{BC}$ for some scalar $k$.
   • For vectors $\mathbf{BC}$ and $\mathbf{CA}$, check if $\mathbf{BC} = k \cdot \mathbf{CA}$ for some scalar $k$.
   • For vectors $\mathbf{CA}$ and $\mathbf{AB}$, check if $\mathbf{CA} = k \cdot \mathbf{AB}$ for some scalar $k$.

   If none of these conditions hold, the vectors form a triangle.

2. Check if the triangle is a right triangle: To determine if the triangle is a right triangle, we check if the vectors satisfy the Pythagorean theorem. Specifically, if $\mathbf{AB}^2 + \mathbf{BC}^2 = \mathbf{CA}^2$ or any of the other permutations.

3. Calculate the area of the triangle: The area $A$ of the triangle formed by vectors can be calculated using the cross product: $\text{Area } = \frac{1}{2} \times \|\mathbf{AB} \times \mathbf{BC}\|$ where $\mathbf{AB} \times \mathbf{BC}$ is the cross product of vectors $\mathbf{AB}$ and $\mathbf{BC}$.

Let's calculate each step.### 1. Collinearity Check:

• The vectors $\mathbf{AB} = (-3, 3, 5)$, $\mathbf{BC} = (-6, -6, -7)$, and $\mathbf{CA} = (9, 3, 2)$ are not collinear. This is confirmed as none of the pairs satisfy the condition for collinearity.

Therefore, these vectors do form a triangle.

2. Right Triangle Check:

• We checked the dot products:
  • $\mathbf{AB} \cdot \mathbf{BC} = -35$
  • $\mathbf{BC} \cdot \mathbf{CA} = -86$
  • $\mathbf{CA} \cdot \mathbf{AB} = -8$

Since none of the dot products is zero, the triangle is not a right triangle.

3. Area of the Triangle:

• The area of the triangle is calculated using the cross product of vectors $\mathbf{AB}$ and $\mathbf{BC}$: $\text{Area} = 31.54 \, \text{square units} \, (\text{approx})$

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you find the cross product of two vectors?
2. What are the conditions for vectors to be collinear?
3. How is the dot product used to determine if vectors are perpendicular?
4. What other methods can be used to find the area of a triangle given its vertices?
5. Can this process be applied to vectors in any dimension?

Tip: Remember, the dot product being zero is a quick way to check for perpendicularity (and thus a right angle) between vectors!