Three Dimensional Geometry Questions

MathematicsThree Dimensional Geometry72 Questions Found

MathematicsShortest Distance Between Two Skew LinesJEE Advanced 2008Moderate

Comprehension Passage

Consider the lines

L_{1} : \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2}

and

L_{2} : \frac{x - 2}{1} = \frac{y + 2}{2} = \frac{z - 3}{3}

Question 1:

The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is

(A)

\frac{- i ^ + 7 j ^ + 7 k ^}{99}

(B)

\frac{- i ^ - 7 j ^ + 5 k ^}{5 3}

(C)

\frac{- i ^ + 7 j ^ + 5 k ^}{5 3}

(D)

\frac{7 i ^ - 7 j ^ - k ^}{99}

Answer:B

Question 2:

The shortest distance between $L_{1}$ and $L_{2}$ is

(A)

(B)

\frac{17}{3}

(C)

\frac{41}{5 3}

(D)

\frac{17}{5 3}

Answer:D

Question 3:

The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines $L_{1}$ and $L_{2}$ is

(A)

\frac{2}{75}

(B)

\frac{7}{75}

(C)

\frac{13}{75}

(D)

\frac{23}{75}

Answer:C

MathematicsEquation of a Line in SpaceJEE Advanced 2006Moderate

View in:English Hindi

Match the following

List-I

(P)

\sum_{i = 1}^{\infty} tan^{- 1} (\frac{1}{2 i ^{2}}) = t

, then

tan t =

(Q)

Sides

a, b, c

of a triangle

A B C

are in AP and

cos θ_{1} = \frac{a}{b + c}

cos θ_{2} = \frac{b}{a + c}

cos θ_{3} = \frac{c}{a + b}

, then

tan^{2} (\frac{θ _{1}}{2}) + tan^{2} (\frac{θ _{3}}{2}) =

(R)

A line is perpendicular to

x + 2 y + 2 z = 0

and passes through

(0, 1, 0)

. The perpendicular distance of this line from the origin is

List-II

(1)

(2)

\frac{5}{3}

(3)

\frac{2}{3}

Answer:P → 1, Q → 3, R → 2

MathematicsEquation of a PlaneJEE Advanced 2006Difficult

View in:English Hindi

Match the following :

List-I

(P)

Two rays

x + y = ∣ a ∣

and

a x - y = 1

intersects each other in the first quadrant in the interval

a \in (a_{0}, \infty)

, the value of

a_{0}

(Q)

Point

(α, β, γ)

lies on the plane

x + y + z = 2

. Let

a = α \hat{i} + β \hat{j} + γ \hat{k}

\hat{k} \times (\hat{k} \times a) = 0

, then

γ =

(R)

\int_{0}^{1} (1 - y^{2}) d y + \int_{1}^{0} (y^{2} - 1) d y

(S)

sin A sin B sin C + cos A cos B = 1

, then the value of

sin C =

List-II

(1)

(2)

4/3

(3)

\int_{0}^{1} 1 - x d x + \int_{- 1}^{0} 1 + x d x

(4)

Answer:P → 4, Q → 1, Q → 3, S → 4

MathematicsEquation of a PlaneJEE Advanced 2013Moderate

View in:English Hindi

Consider the lines $L_{1} : \frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, L_{2} : \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2}$ and the planes $P_{1} : 7 x + y + 2 z = 3, P_{2} : 3 x + 5 y - 6 z = 4$ . Let $a x + b y + cz = d$ be the equation of the plane passing through the point of intersection of lines $L_{1}$ and $L_{2}$ , and perpendicular to planes $P_{1}$ and $P_{2}$ . Match List I with List II:

List-I

(P)

(Q)

(R)

(S)

List-II

(1)

(2)

-3

(3)

(4)

-2

Answer:P → 3, Q → 2, R → 4, S → 1

MathematicsEquation of a PlaneJEE Advanced 2004Moderate

View in:English Hindi

$P_{1}$ and $P_{2}$ are planes passing through origin. $L_{1}$ and $L_{2}$ are two line on $P_{1}$ and $P_{2}$ respectively such that their intersection is origin. Show that there exists points $A, B, C$ whose permutation $A^{'}, B^{'}, C^{'}$ can be chosen such that (i) $A$ is on $L_{1}, B$ on $P_{1}$ but not on $L_{1}$ and $C$ not on $P_{1}$ (ii) $A^{'}$ is on $L_{2}, B^{'}$ on $P_{2}$ but not on $L_{2}$ and $C^{'}$ not on $P_{2}$ .

MathematicsEquation of a PlaneJEE Advanced 2004Moderate

View in:English Hindi

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.

MathematicsEquation of a Line in SpaceJEE Advanced 1996Difficult

View in:English Hindi

The position vectors of the vertices $A, B$ and $C$ of a tetrahedron $A B C D$ are $\hat{i} + \hat{j} + \hat{k}, \hat{i}$ and $3 \hat{i}$ , respectively. The altitude from vertex $D$ to the opposite face $A B C$ meets the median line through $A$ of the triangle $A B C$ at a point $E$ . If the length of the side $A D$ is $4$ and the volume of the tetrahedron is $\frac{2 2}{3}$ , find the position vector of the point $E$ for all its possible positions.

Answer:(-1, 3, 3) or (3, -1, -1)

MathematicsEquation of a PlaneJEE Advanced 1996Moderate

View in:English Hindi

A nonzero vector $a$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i}, \hat{i} + \hat{j}$ and the plane determined by the vectors $\hat{i} - \hat{j}, \hat{i} + \hat{k}$ . The angle between $a$ and the vector $\hat{i} - 2 \hat{j} + 2 \hat{k}$ is .........

Answer:

\frac{π}{4}

\frac{3 π}{4}

MathematicsEquation of a PlaneJEE Advanced 2003Moderate

View in:English Hindi

(i) Find the equation of the plane passing through the points $(2, 1, 0), (5, 0, 1)$ and $(4, 1, 1)$ . (ii) If $P$ is the point $(2, 1, 6)$ then find the point $Q$ such that $P Q$ is perpendicular to the plane in (i) and the midpoint of $P Q$ lies on it.

Answer:(i)

x + y - 2 z = 3

(ii)

Q (6, 5, - 2)

MathematicsEquation of a PlaneJEE Advanced 2004Moderate

View in:English Hindi

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.

Answer:

x + y + z = 3

;

\frac{9}{2}

cubic units

...8

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The unit vector perpendicular to both L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines L1​ and L2​ is

Match the following

List-I

List-II

Match the following :

List-I

List-II

List-I

List-II

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The unit vector perpendicular to both L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines L1​ and L2​ is

Match the following

List-I

List-II

Match the following :

List-I

List-II

List-I

List-II

The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is

The shortest distance between $L_{1}$ and $L_{2}$ is

The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines $L_{1}$ and $L_{2}$ is

The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is

The shortest distance between $L_{1}$ and $L_{2}$ is

The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines $L_{1}$ and $L_{2}$ is