Answer :
Sure, let's go through this step-by-step to determine the relationships between the lines \(PQ\) and \(RS\) in both cases.
### Case (a): \(P(1,2,3)\), \(Q(4,5,6)\), \(R(-2,3,5)\), \(S(4,9,11)\)
1. Calculate Direction Vectors:
- For line \(PQ\):
[tex]\[ \vec{PQ} = Q - P = (4-1, 5-2, 6-3) = (3, 3, 3) \][/tex]
- For line \(RS\):
[tex]\[ \vec{RS} = S - R = (4-(-2), 9-3, 11-5) = (6, 6, 6) \][/tex]
2. Cross Product:
To check if the vectors are parallel, we compute the cross product:
[tex]\[ \vec{PQ} \times \vec{RS} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 3 & 3 \\ 6 & 6 & 6 \end{matrix} \right| = (0, 0, 0) \][/tex]
Since the cross product is zero, the direction vectors are parallel.
3. Dot Product:
To check if they are perpendicular, we compute the dot product:
[tex]\[ \vec{PQ} \cdot \vec{RS} = 3 \cdot 6 + 3 \cdot 6 + 3 \cdot 6 = 18 + 18 + 18 = 54 \][/tex]
The dot product is not zero, so the vectors are not perpendicular.
Conclusion for Case (a): The lines \(PQ\) and \(RS\) are parallel.
### Case (b): \(P(3,-1,-3)\), \(Q(2,-3,1)\), \(R(3,-2,5)\), \(S(-1,-2,1)\)
1. Calculate Direction Vectors:
- For line \(PQ\):
[tex]\[ \vec{PQ} = Q - P = (2-3, -3-(-1), 1-(-3)) = (-1, -2, 4) \][/tex]
- For line \(RS\):
[tex]\[ \vec{RS} = S - R = (-1-3, -2-(-2), 1-5) = (-4, 0, -4) \][/tex]
2. Cross Product:
To check if the vectors are parallel, we compute the cross product:
[tex]\[ \vec{PQ} \times \vec{RS} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & -2 & 4 \\ -4 & 0 & -4 \end{matrix} \right| = (8, -20, -8) \][/tex]
Since the cross product is not zero, the direction vectors are not parallel.
3. Dot Product:
To check if they are perpendicular, we compute the dot product:
[tex]\[ \vec{PQ} \cdot \vec{RS} = (-1) \cdot (-4) + (-2) \cdot 0 + 4 \cdot (-4) = 4 + 0 - 16 = -12 \][/tex]
The dot product is not zero, so the vectors are not perpendicular.
Conclusion for Case (b): The lines \(PQ\) and \(RS\) intersect.
### Summary:
- Case (a): The lines \(PQ\) and \(RS\) are parallel.
- Case (b): The lines [tex]\(PQ\)[/tex] and [tex]\(RS\)[/tex] intersect but are not perpendicular.
### Case (a): \(P(1,2,3)\), \(Q(4,5,6)\), \(R(-2,3,5)\), \(S(4,9,11)\)
1. Calculate Direction Vectors:
- For line \(PQ\):
[tex]\[ \vec{PQ} = Q - P = (4-1, 5-2, 6-3) = (3, 3, 3) \][/tex]
- For line \(RS\):
[tex]\[ \vec{RS} = S - R = (4-(-2), 9-3, 11-5) = (6, 6, 6) \][/tex]
2. Cross Product:
To check if the vectors are parallel, we compute the cross product:
[tex]\[ \vec{PQ} \times \vec{RS} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 3 & 3 \\ 6 & 6 & 6 \end{matrix} \right| = (0, 0, 0) \][/tex]
Since the cross product is zero, the direction vectors are parallel.
3. Dot Product:
To check if they are perpendicular, we compute the dot product:
[tex]\[ \vec{PQ} \cdot \vec{RS} = 3 \cdot 6 + 3 \cdot 6 + 3 \cdot 6 = 18 + 18 + 18 = 54 \][/tex]
The dot product is not zero, so the vectors are not perpendicular.
Conclusion for Case (a): The lines \(PQ\) and \(RS\) are parallel.
### Case (b): \(P(3,-1,-3)\), \(Q(2,-3,1)\), \(R(3,-2,5)\), \(S(-1,-2,1)\)
1. Calculate Direction Vectors:
- For line \(PQ\):
[tex]\[ \vec{PQ} = Q - P = (2-3, -3-(-1), 1-(-3)) = (-1, -2, 4) \][/tex]
- For line \(RS\):
[tex]\[ \vec{RS} = S - R = (-1-3, -2-(-2), 1-5) = (-4, 0, -4) \][/tex]
2. Cross Product:
To check if the vectors are parallel, we compute the cross product:
[tex]\[ \vec{PQ} \times \vec{RS} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & -2 & 4 \\ -4 & 0 & -4 \end{matrix} \right| = (8, -20, -8) \][/tex]
Since the cross product is not zero, the direction vectors are not parallel.
3. Dot Product:
To check if they are perpendicular, we compute the dot product:
[tex]\[ \vec{PQ} \cdot \vec{RS} = (-1) \cdot (-4) + (-2) \cdot 0 + 4 \cdot (-4) = 4 + 0 - 16 = -12 \][/tex]
The dot product is not zero, so the vectors are not perpendicular.
Conclusion for Case (b): The lines \(PQ\) and \(RS\) intersect.
### Summary:
- Case (a): The lines \(PQ\) and \(RS\) are parallel.
- Case (b): The lines [tex]\(PQ\)[/tex] and [tex]\(RS\)[/tex] intersect but are not perpendicular.
