. Find the volume of the parallelepiped determined by the vectors a = i + j, b = j + k, and c=i+j+k 6. Given that P(3,0,1),Q(−1,2,5),R(5,1,−1), and S(0,4,2). Find the volume of the paral- lelepiped with adjacent edges PQ,PR, and PS. 7. Given that P (1, 1, 1), Q(2, 1, 3), and R(3, −1, 1) are vertices of a triangle. (a) Find the area of the triangle determined by the points P, Q, and R. (b) Find a unit vector perpendicular to plane PQR. 8. Leta=i+2j−k,b=−i+j+k,andc=i+k.Whichvectors,ifany,are (a) perpendicular? (b) Parallel? Give reasons for your answers. 5. Find the volume of the parallelepiped determined by the vectors a = i + j, b = j + k, and c=i+j+k 6. Given that P(3,0,1),Q(−1,2,5),R(5,1,−1), and S(0,4,2). Find the volume of the paral- lelepiped with adjacent edges PQ,PR, and PS. 7. Given that P (1, 1, 1), Q(2, 1, 3), and R(3, −1, 1) are vertices of a triangle. (a) Find the area of the triangle determined by the points P, Q, and R. (b) Find a unit vector perpendicular to plane PQR. 8. Leta=i+2j−k,b=−i+j+k,andc=i+k.Whichvectors,ifany,are (a) perpendicular? (b) Parallel? Give reasons for your answers.
The Correct Answer and Explanation is :
Let’s address each problem step by step:
5. Volume of the Parallelepiped Determined by Vectors a, b, and c
Given vectors:
- a = i + j = (1, 1, 0)
- b = j + k = (0, 1, 1)
- c = i + j + k = (1, 1, 1)
The volume ( V ) of a parallelepiped formed by vectors a, b, and c is given by the absolute value of the scalar triple product:
[ V = |a \cdot (b \times c)| ]
First, compute the cross product b × c:
[
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
0 & 1 & 1 \
1 & 1 & 1
\end{vmatrix}
= \mathbf{i}(1 \cdot 1 – 1 \cdot 1) – \mathbf{j}(0 \cdot 1 – 1 \cdot 1) + \mathbf{k}(0 \cdot 1 – 1 \cdot 1)
= \mathbf{i}(0) – \mathbf{j}(-1) + \mathbf{k}(-1)
= \mathbf{j} – \mathbf{k}
]
So, b × c = j – k = (0, 1, -1).
Next, compute the dot product a · (b × c):
[
(1, 1, 0) \cdot (0, 1, -1) = 1 \cdot 0 + 1 \cdot 1 + 0 \cdot (-1) = 0 + 1 + 0 = 1
]
Therefore, the volume ( V ) is:
[ V = |1| = 1 ]
The volume of the parallelepiped is 1 cubic unit.
6. Volume of the Parallelepiped with Adjacent Edges PQ, PR, and PS
Given points:
- P(3, 0, 1)
- Q(-1, 2, 5)
- R(5, 1, -1)
- S(0, 4, 2)
First, determine vectors PQ, PR, and PS:
- PQ = Q – P = (-1 – 3, 2 – 0, 5 – 1) = (-4, 2, 4)
- PR = R – P = (5 – 3, 1 – 0, -1 – 1) = (2, 1, -2)
- PS = S – P = (0 – 3, 4 – 0, 2 – 1) = (-3, 4, 1)
The volume ( V ) is given by the absolute value of the scalar triple product:
[ V = |PQ \cdot (PR \times PS)| ]
Compute the cross product PR × PS:
[
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
2 & 1 & -2 \
-3 & 4 & 1
\end{vmatrix}
= \mathbf{i}(1 \cdot 1 – (-2) \cdot 4) – \mathbf{j}(2 \cdot 1 – (-2) \cdot (-3)) + \mathbf{k}(2 \cdot 4 – 1 \cdot (-3))
= \mathbf{i}(1 + 8) – \mathbf{j}(2 – 6) + \mathbf{k}(8 + 3)
= 9\mathbf{i} + 4\mathbf{j} + 11\mathbf{k}
]
So, PR × PS = (9, 4, 11).
Next, compute the dot product PQ · (PR × PS):
[
(-4, 2, 4) \cdot (9, 4, 11) = (-4) \cdot 9 + 2 \cdot 4 + 4 \cdot 11 = -36 + 8 + 44 = 16
]
Therefore, the volume ( V ) is:
[ V = |16| = 16 ]
The volume of the parallelepiped is 16 cubic units.
7. Triangle with Vertices P, Q, and R
Given points:
- P(1, 1, 1)
- Q(2, 1, 3)
- R(3, -1, 1)
(a) Area of the Triangle
First, determine vectors PQ and PR:
- PQ = Q – P = (2 – 1, 1 – 1, 3 – 1) = (1, 0, 2)
- PR = R – P = (3 – 1, -1 – 1, 1 – 1) = (2, -2, 0)
Compute the cross product PQ × PR:
[
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
1 & 0 & 2 \
2 & -2 & 0
\end{vmatrix}
= \mathbf{i}(0 \cdot 0 – 2 \cdot (-2)) – \mathbf{j}(1 \cdot 0 – 2 \