To determine if three points lie on a straight line in three-dimensional space, we calculate the vectors between pairs of points and check if they are scalar multiples of each other. If the vectors are scalar multiples, the points are collinear (on a straight line). Here’s the process for each set of points:

1. Points A(0,0,0), B(7,−3,4), and C(−28,12,−16)

• Compute vectors AB and AC:
  • $\mathbf{AB} = (7 - 0, -3 - 0, 4 - 0) = (7, -3, 4)$
  • $\mathbf{AC} = (-28 - 0, 12 - 0, -16 - 0) = (-28, 12, -16)$
• Check if $\mathbf{AC} = k \cdot \mathbf{AB}$:
  • $-28 = 7k, \; 12 = -3k, \; -16 = 4k$
  • Solving: $k = -4$ for all components.

Answer: Yes, the points lie on a straight line.

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2. Points D(7,−3,4), E(8,−1,7), and F(12,7,19)

• Compute vectors DE and DF:
  • $\mathbf{DE} = (8 - 7, -1 - (-3), 7 - 4) = (1, 2, 3)$
  • $\mathbf{DF} = (12 - 7, 7 - (-3), 19 - 4) = (5, 10, 15)$
• Check if $\mathbf{DF} = k \cdot \mathbf{DE}$:
  • $5 = 1k, \; 10 = 2k, \; 15 = 3k$
  • Solving: $k = 5$ for all components.

Answer: Yes, the points lie on a straight line.

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3. Points G(−1,0,1), H(3,7,8), and (incomplete)

You have not provided the third point for this set. Please share the coordinates of the third point to proceed.

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Would you like further clarification or a breakdown of any part? Here are some follow-up questions:

1. What is the importance of collinearity in geometry and physics?
2. How are vector equations useful in determining collinearity in higher dimensions?
3. Could scalar multiplication be applied to determine parallel vectors in real-world scenarios?
4. What happens if the scalar multiplier $k$ does not satisfy all components?
5. Can you provide additional sets of points to test collinearity?

Tip: Always double-check your vector calculations, as a small error in one component could change the result!