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Find the shortest distance between any two opposite edges of a tetrahedron formed by the planes $y + z = 0, x + z = 0, x + y = 0, x + y + z = \sqrt{3} a$

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Important Questions on Three Dimensional Geometry

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 3

Equation of plane which passes through the point of intersection of lines

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

and

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

and at greatest distance from the point

(0, 0, 0)

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 4

The non zero value of

a

for which the line

2 x - y + 3 z + 4 = 0 = a x + y - z + 2

and

x - 3 y + z = 0 = x + 2 y + z + 1

are co-planar is

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 6

A line

L_{1}

having direction ratios

1, 0, 1

lies on

x z

-plane is rotated about

z

-axis by an angle

90^{°}

. Now the new position of

L_{1}

L_{2}

. The angle between

L_{1} & L_{2}

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 8

Given three point on

x - y

plane as

O (0, 0), A (1, 0) & B (- 1, 0)

. Point

P

is moving on the plane satisfying the condition

(\vec{P A} \cdot \vec{P B}) + 3 (\vec{O A} \cdot \vec{O B}) = 0

. If the maximum & minimum values of

|\vec{P A}| |\vec{P B}|

are

M & m

respectively then the value of

M^{2} + m^{2}

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 9

\vec{r}

represents the position vector of point

R

in which the line

A B

cuts the plane

C D E,

where the position vectors of points

A, B, C, D, E

are respectively

\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}, \vec{c} = - 4 \vec{j} + 4 \hat{k}, \vec{d} = 2 \hat{i} - 2 \hat{j} + 2 \hat{k}

and

\vec{e} = 4 \hat{i} + \hat{j} + 2 \hat{k},

then

{\vec{r}}^{2}

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 10

Line

L_{1}

is parallel to vector

\vec{α} = - 3 \hat{i} + 2 \hat{j} + 4 \hat{k}

and passes through a point

A (7, 6, 2)

and line

L_{2}

is parallel to a vector

\vec{β} = 2 \hat{i} + \hat{j} + 3 \hat{k}

and passes through a point

B (5, 3, 4) .

Now, a line

L_{3}

parallel to a vector

\vec{r} = 2 \hat{i} - 2 \hat{ȷ} - \hat{k}

intersects the lines

L_{1}

and

L_{2}

at points

C

and

D

respectively, then

|4 \vec{C D}|

is equal to ?

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 13

The value of

\sec^{3} θ,

where

θ

is the acute angle between the plane faces of a regular tetrahedron, is

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 14

R

and

r

are the circum-radius and in-radius of a regular tetrahedron respectively in terms of the length

k

of each edge. If

R^{2} + r^{2} = \frac{p}{q} k^{2},

where

p, q \in I

, then absolute minimum value of

p + q

Find the shortest distance between any two opposite edges of a tetrahedron formed by the planes y+z=0,x+z=0,x+y=0,x+y+z=3a

Important Questions on Three Dimensional Geometry

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Find the shortest distance between any two opposite edges of a tetrahedron formed by the planes $y + z = 0, x + z = 0, x + y = 0, x + y + z = \sqrt{3} a$