A parallelopiped 'S' has base points A,...

MathematicsEquation of a PlaneJEE Advanced 2004Moderate

A parallelopiped 'S' has base points $A, B, C$ and $D$ and upper face points $A^{'}, B^{'}, C^{'}$ and $D^{'}$ . This parallelopiped is compressed by upper face $A^{'} B^{'} C^{'} D^{'}$ to form a new parallelopiped 'T' having upper face points $A^{''}, B^{''}, C^{''}$ and $D^{''}$ . Volume of parallelopiped $T$ is 90 percent of the volume of parallelopiped $S$ . Prove that the locus of $A^{''}$ , is a plane.

Visualized Solution (Hindi)

Visualizing Parallelepiped .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Consider parallelepiped $S$ with base $A B C D$ and top face $A^{'} B^{'} C^{'} D^{'}$ .
Let the area of the base $A B C D$ be $A_{ba se}$ .
Let the height of $S$ be $h$ .

Volume of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Volume of a parallelepiped is given by:
Volume = Area of base $\times$ Height
For $S$ : $V_{S} = A_{ba se} \times h$

Defining the Base Plane

Let the equation of the base plane $A B C D$ be:
Plane Equation: $a x + b y + cz + d = 0$

Compression to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T

Parallelepiped $S$ is compressed to form $T$ .
Base $A B C D$ remains the same.
New upper face is $A^{''} B^{''} C^{''} D^{''}$ .

Volume Relation

Given: $V_{T} = 90% of V_{S}$
Relation: $V_{T} = 0.9 V_{S}$

Finding New Height .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } h^{'}

$A_{ba se} \times h^{'} = 0.9 (A_{ba se} \times h)$
New Height: $h^{'} = 0.9 h$

Distance Formula for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A^{''}

Let $A^{''}$ be $(α, β, γ)$ .
Height $h^{'}$ is the distance from $A^{''}$ to plane $a x + b y + cz + d = 0$ .
Distance Formula: $h^{'} = \frac{∣ a α + b β + c γ + d ∣}{a ^{2} + b ^{2} + c ^{2}}$

Equating the Heights

Equating the two expressions for $h^{'}$ :
$\frac{∣ a α + b β + c γ + d ∣}{a ^{2} + b ^{2} + c ^{2}} = 0.9 h$

Finding the Locus

$a α + b β + c γ + d = \pm 0.9 h a^{2} + b^{2} + c^{2}$
Replacing $(α, β, γ)$ with $(x, y, z)$ :
Locus: $a x + b y + cz + (d \mp 0.9 h a^{2} + b^{2} + c^{2}) = 0$

Conclusion: Parallel Plane

The locus equation is of the form $A x + B y + C z + D^{'} = 0$ .
This represents a plane parallel to the base plane $A B C D$ .
Key Takeaway: A constant volume ratio for a fixed base implies a constant height, resulting in a parallel plane locus.

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Conceptually Similar Problems

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The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\overset{a}{^}, \hat{b}, \overset{c}{^}$ such that $\overset{a}{^} \cdot \hat{b} = \hat{b} \cdot \overset{c}{^} = \overset{c}{^} \cdot \overset{a}{^} = \frac{1}{2}$ . Then, the volume of the parallelopiped is

(A)

\frac{1}{2}

(B)

\frac{1}{2 2}

(C)

\frac{3}{2}

(D)

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Answer:A

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The volume of the parallelopiped whose sides are given by $O A = 2 \hat{i} - 2 \hat{j}, O B = \hat{i} + \hat{j} - \hat{k}, O C = 3 \hat{i} - \hat{k}$ , is

(A)

4/13

(B)

(C)

2/7

(D)

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The value of 'a' so that the volume of parallelopiped formed by $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

(A)

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(B)

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(C)

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;

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The end $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ be completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $A B$ is $x^{2/3} + y^{2/3} = c^{2/3}$ .

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Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

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If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

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Match List I with List II:

List-I

(P)

Volume of parallelepiped determined by vectors

a, b

and

c

is 2. Then the volume of the parallelepiped determined by vectors

2 (a \times b), 3 (b \times c)

and

2 (c \times a)

(Q)

Volume of parallelepiped determined by vectors

a, b

and

c

is 5. Then the volume of the parallelepiped determined by vectors

3 (a + b), 3 (b + c)

and

2 (c + a)

(R)

Area of a triangle with adjacent sides determined by vectors

a

and

b

is 20. Then the area of the triangle with adjacent sides determined by vectors

(2 a + 3 b)

and

(a - b)

(S)

Area of a parallelogram with adjacent sides determined by vectors

a

and

b

is 30. Then the area of the parallelogram with adjacent sides determined by vectors

(a + b)

and

a

List-II

(1)

100

(2)

(3)

(4)

Answer:P → 3, Q → 4, R → 1, S → 2

Visualized Solution (Hindi)

Visualizing Parallelepiped $S$

Volume of $S$

Defining the Base Plane

Compression to $T$

Volume Relation

Finding New Height $h^{'}$

Distance Formula for $A^{''}$

Equating the Heights

Finding the Locus

Conclusion: Parallel Plane

Conceptually Similar Problems

The volume of the parallelopiped whose sides are given by $O A = 2 \hat{i} - 2 \hat{j}, O B = \hat{i} + \hat{j} - \hat{k}, O C = 3 \hat{i} - \hat{k}$ , is

The value of 'a' so that the volume of parallelopiped formed by $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

Find the equation of plane passing through $(1, 1, 1)$ & parallel to the lines $L_{1}, L_{2}$ having direction ratios $(1, 0, - 1), (1, - 1, 0)$ . Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.

The end $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ be completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $A B$ is $x^{2/3} + y^{2/3} = c^{2/3}$ .

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

Match List I with List II:

List-I

List-II

The volume of the parallelopiped whose sides are given by $O A = 2 \hat{i} - 2 \hat{j}, O B = \hat{i} + \hat{j} - \hat{k}, O C = 3 \hat{i} - \hat{k}$ , is

The value of 'a' so that the volume of parallelopiped formed by $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

Find the equation of plane passing through $(1, 1, 1)$ & parallel to the lines $L_{1}, L_{2}$ having direction ratios $(1, 0, - 1), (1, - 1, 0)$ . Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.

The end $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ be completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $A B$ is $x^{2/3} + y^{2/3} = c^{2/3}$ .

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

Match List I with List II:

List-I

List-II

Visualized Solution (Hindi)

Visualizing Parallelepiped .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Volume of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Defining the Base Plane

Compression to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T

Volume Relation

Finding New Height .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } h′

Distance Formula for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A′′

Equating the Heights

Finding the Locus

Conclusion: Parallel Plane

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The volume of the parallelopiped whose sides are given by OA=2i^−2j^​,OB=i^+j^​−k^,OC=3i^−k^, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 'a' so that the volume of parallelopiped formed by i^+aj^​+k^, j^​+ak^ and ai^+k^ becomes minimum is

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors b,c,d are not coplanar, then prove that the vector (a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c) is parallel to a.

Match List I with List II:

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The volume of the parallelopiped whose sides are given by OA=2i^−2j^​,OB=i^+j^​−k^,OC=3i^−k^, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 'a' so that the volume of parallelopiped formed by i^+aj^​+k^, j^​+ak^ and ai^+k^ becomes minimum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vectors b,c,d are not coplanar, then prove that the vector (a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c) is parallel to a.

Match List I with List II:

List-I

List-II

Visualizing Parallelepiped $S$

Volume of $S$

Compression to $T$

Finding New Height $h^{'}$

Distance Formula for $A^{''}$

The volume of the parallelopiped whose sides are given by $O A = 2 \hat{i} - 2 \hat{j}, O B = \hat{i} + \hat{j} - \hat{k}, O C = 3 \hat{i} - \hat{k}$ , is

The value of 'a' so that the volume of parallelopiped formed by $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .

The volume of the parallelopiped whose sides are given by $O A = 2 \hat{i} - 2 \hat{j}, O B = \hat{i} + \hat{j} - \hat{k}, O C = 3 \hat{i} - \hat{k}$ , is

The value of 'a' so that the volume of parallelopiped formed by $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

If the vectors $b, c, d$ are not coplanar, then prove that the vector $(a \times b) \times (c \times d) + (a \times c) \times (d \times b) + (a \times d) \times (b \times c)$ is parallel to $a$ .