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QUESTION BANK mathematics ( 3D GEOMETERY)
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THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER 3D GEORMETERY ALONG WITH THE ANSWER KEY AT THE END
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Three DimensionalGeometry
QUICK RECAP
DIRECTION COSINES AND DIRECTION makes with the positive directions of the co-
RATIOS OF A LINE ordinate axes OX, OY and OZ respectively, then
8 If a, b, g are the angles which a vector OP cos a, cos b, cos g are known as the direction
,
cosines of OP and are generally denoted by 8 Cartesian form : If q is the angle between the
l, m and n respectively i.e. l = cos a, m = cos b, x − x1 y − y1 z − z1
lines l1 : = = and
n = cos g. a1 b1 c1
Let l, m, n be the direction cosines of a vector x − x 2 y − y2 z − z 2
l2 : = =
r and a, b, c be three numbers such that a2 b2 c2
l m n a1a2 + b1b2 + c1c2
= = .
a b c then cos θ =
a12 + b12 + c12 . a22 + b22 + c22
Then a, b, c are known as direction ratios or
direction numbers of vector r . Note : The lines are perpendicular to each
Note : other if a1a2 + b1b2 + c1c2 = 0 and parallel if
(i) l2 + m2 + n2 = 1 i.e., cos2 a + cos2b + cos2g = 1 a1 b1 c1
= = .
(ii) Direction ratios of a line joining a2 b2 c2
P(x1, y1, z1) and Q(x2, y2, z2) are x2 – x1,
y2 – y1, z2 – z1 and direction cosines are SHORTEST DISTANCE BETWEEN TWO LINES
x −x y −y z −z
± 2
1 , ± 2
1 , ± 2
1 . 8 Distance between skew lines
| PQ | | PQ | | PQ | X Vector form : Let l1 and l2 be two lines whose
equations are r = a1 + λ b1 and
EQUATION OF A LINE IN SPACE
r = a2 + µ b2 respectively. Let PQ be the
8 Vector equation of a line passing through
shortest distance vector between l1 and l2.
a given point with position vector a and
parallel to a given vector b is r = a + λb , Then,
l ∈ R. (b1 × b2 ) ⋅ (a2 − a1 )
PQ =
8 Cartesian equation of a line passing through a | b1 × b2 |
point A(x1, y1, z1) having position vector a and X Cartesian form : Let two skew lines be
in the direction of a vector having a1, b1, c1 as x − x1 y − y1 z − z1
x − x1 y − y1 z − z1 l1 : = = and
direction ratios, is = = a1 b1 c1
a1 b1 c1
x − x 2 y − y2 z − z 2
l2 : = = .
Equation of Line Passing Through Two Given a2 b2 c2
Points The shortest distance between l1 and l2 is given
8 Let a and b be position vectors of two points by
A(x1, y1, z1) and B(x2, y2, z2) respectively. Let x2 − x1 y2 − y1 z2 − z1
r be the position vector of P(x, y, z). Then, a1 b1 c1
equation of line is
a2 b2 c2
X Vector form: r = a + λ(b − a ), λ ∈ R .
2 2 2
(b1c2 − b2 c1 ) + (c1a2 − a1c2 ) + (a1b2 − a2b1 )
x − x1 y − y1 z − z1
X Cartesian form : = =
x2 − x1 y2 − y1 z2 − z1 Note : If the shortest distance between two
lines is zero, then the lines are intersecting.
ANGLE BETWEEN TWO LINES 8 Distance between parallel lines
8 Vector form : If q is the angle between the X The shortest distance between two parallel
lines r = a1 + λb1 and r = a2 + µ b2 , then lines r = a1 + λb and r = a2 + λb is
b1 . b2 b × (a2 − a1 )
cos θ = d= .
| b1 | | b2 | |b |
,
PLANE the planes r ⋅ n1 = d1 and r ⋅ n2 = d2 is
8 A plane is a surface such that if any two points (r ⋅ n1 − d1 ) + λ (r ⋅ n2 − d2 ) = 0 or we write
are taken on it, the line segment joining them
r ⋅ (n1 + λn2 ) = d1 + λd2 .
lies completely on the surface.
X Cartesian form : The equation of a plane
8 Equation of a plane in normal form passing through the intersection of planes a1x
X Vector form : When a unit vector (n ) is + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0
normal to the plane and d is the perpendicular is (a1x + b1y + c1z + d1) + l (a2x + b2y + c2z + d2) = 0,
distance from the origin, then the equation where l is a constant.
of plane is r . n
= d.
COPLANARITY OF TWO LINES
X Cartesian form : If unit vector n = li + mj + nk ,
where l, m, n are direction cosines and d, the 8 Vector form : Let r = a1 + λb1 and
perpendicular distance from origin to plane, r = a2 + µb2 be two lines, then these lines
then equation of the plane is lx + my + nz = d.
are coplanar iff [a2 − a1 b1 b2 ] = 0
8 Equation of a plane perpendicular to a given 8 Cartesian form :
vector and passing through a given point x − x1 y − y1 z − z1
Equation of a plane passing through a point Let = = and
a1 b1 c1
with position vector a = x
i + y
j + z k
and
1 1 1 x − x 2 y − y2 z − z 2
= = are two lines, then
perpendicular to the vector n = Ai
+ B
j + Ck
is a2 b2 c2
X Vector form : (r − a ) . n = 0 or r . n = a . n these lines are coplanar iff
X Cartesian form : x2 − x1 y2 − y1 z 2 − z1
A(x – x1) + B(y – y1) + C(z – z1) = 0. a1 b1 c1 = 0
8 Equation of a plane passing through three a2 b2 c2
non-collinear points : If a plane passes
through points A(x1, y1, z1), B(x2, y2, z2) ANGLE BETWEEN TWO PLANES
and C(x3, y3, z3) having position vectors
8 Vector form : Let r . n1 = d1 and r . n2 = d2
a , b and c respectively, and let P(x, y, z) be be two planes making an angle q with each
any point on the plane having position vector other. Then q is the angle between their
r , then equation of plane is normals n1 and n2 respectively.
X Vector form : (r − a ) . [(b − a ) × (c − a )] = 0
n1 . n2
X Cartesian form : Therefore, cos θ =
| n1 | | n2 |
x − x1 y − y1 z − z1
x2 − x1 y2 − y1 z 2 − z1 = 0 8 Cartesian form : If a1x + b1y + c1z + d1 = 0
x3 − x1 y3 − y1 z3 − z1 and a2x + b2y + c2z + d2 = 0 be two planes and
q is the angle between them, where a1, b1, c1
8 Intercept form of the equation of a plane : and a2, b2, c2 are direction ratios of normals
If a plane makes intercepts a, b and c on x,
to the planes, then
y and z-axes respectively, then equation of
x y z a1a2 + b1b2 + c1c2
plane in intercept form is + + = 1. cos θ =
a b c a12 + b12 + c12 a22 + b22 + c22
8 Equation of a plane passing through the
intersection of two planes Note : The planes are perpendicular to each
X Vector form : The equation of a plane other if n1 . n2 = 0 and parallel if n1 = λn2 ,
passing through the intersection of where l is a scalar.
, DISTANCE OF A POINT FROM A PLANE
b .n
r . n = d , then cos θ = .
8 Vector form : The length of the perpendicular |b ||n |
from a point having position vector a to the
| a ⋅n−d | So, the angle f between the line and the plane
plane r . n = d is given by . is 90° – q i.e., sin(90° – q) = cos q
|n |
8 Cartesian form : The length of the b .n
perpendicular from a point P(x1, y1, z1) to ⇒ sin φ =
|b ||n |
the plane ax + by + cz + d = 0 is given by
ax1 + by1 + cz1 + d 8 Cartesian form : Let a1, a2, a3 are the direction
. ratios of line and lx + my + nz + d = 0 be the
a 2 + b2 + c 2
equation of plane. If q is the angle between
ANGLE BETWEEN A LINE AND A PLANE the line and plane, then
a1l + a2m + a3n
8 Vector form : If q is the angle between a sin θ =
a12 + a22 + a32 l 2 + m2 + n 2
line r = a + λb and normal to the plane
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