1.If P(x1, y1, z1) and Q(x2, y2, z2) are two points in space, then
2.The distance of a point P(x1, y1, z1) from the origin 0 is given by
3.The coordinates of a point R which divides the line joining the points P(x1, y1, z1) and Q(x2, y2, z2 ) internally in the ratio m : n are
4.The coordinates of a point R which divides the line joining the points P(x1, y1, z1) and Q(x2 y2, z2) externally in the ratio m : n are
5.Let P(x1, y1 z1) and Q(x2 y2, z2) be two points in space. The coordinates of the midpoint of PQ are
6.Let P(x1, y1 z1), Q(x2 y2, z2) and R(x3 y3, z3) be three vertices of the triangle.
Hence, the centroid
7.The projection of the line joining points P(x1, y1, z1) and Q(x2, y2, z2) to the line with direction cosines, I, m and n is
Top Concepts.
1.The angles α, β and γ which a directed line L, through the origin, makes with the x , y and z axes, respectively, are called direction angles.
If the direction of line L is reversed, then the direction angles will
2.If a directed line L passes through the origin and makes angles α , β and γ with the x, y and z axes respectively, then
λ = cos α , m = cos β and n = cos γ are called direction cosines of line L.
3.For a given line to have a unique set of direction cosines, a directed line is used.
4.The direction cosines of the directed line which does not pass through the origin can be obtained by drawing a line parallel to it and passing through the origin.
5.Any three numbers which are proportional to the direction cosines of the line are called direction ratios. If λ, m and n are the direction cosines and a, b and c are the direction ratios of a line, then λ = ka, m = kb and n = kc, where k is any non-zero real number.
6.For any line, there are an infinite number of direction ratios.
7.Direction ratios of the line joining P(x1, y1, z1) and Q(x2, y2, z2) may be taken as
8.Direction cosines of the x-axis are cos0, cos90, cos90, i.e., 1, 0, 0.
Similarly, the direction cosines of the y-axis are 0, 1, 0 and the z-axis are 0, 0, 1, respectively.
9.A line is uniquely determined if
1.It passes through a given point and has given direction ratios
OR
2.It passes through two given points.
10.Two lines with direction ratios a1, a2, a3 and b1, b2, b3, respectively, are perpendicular if
a1b1 + a2b2 + a3b3 = 0
11.Two lines with direction ratios a1, a2, a3 and b1, b2, b3, respectively, are parallel if
12.The lines which are neither intersecting nor parallel are called as skew lines. Skew lines are non- coplanar, i.e., they do not belong to the same 2D plane.
GE and DB are skew lines.
13.The angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.
14.If two lines in space are intersecting, then the shortest distance between them is zero.
15.If two lines in space are parallel, then the shortest distance between them is the perpendicular distance.
16.The normal vector, often simply called the ‘normal’ to a surface, is a vector perpendicular to a surface.
17.If the three points are collinear, then the line containing those three points can be part of many planes.
18.The angle between two planes is defined as the angle between their normals.
19.If the planes A1x + B1y + C1z + D1 = 0 and A2 x + B2y + C2z + D2 = 0 are perpendicular to each other, then A1A2 + B1B2 + C1C2 = 0
If the planes A1x+ B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are parallel, then
20. The angle between a line and a plane is the complement of the angle between the line and the normal to the plane.
21.The distance of a point from a plane is the length of the unique line from the point to the plane which is perpendicular to the plane.
Top Formulae
.
1.Direction cosines of the line L are connected by the relation l2 + m2 + n2 = 1.
2.If a, b, and c are the direction ratios of a line, and l, m, and n are its direction cosines, then
3.The direction cosines of the line joining
P( x1, y1, z1) and Q( x2, y2, z2) are
where
4.Vector equation of a line which passes through the given point whose position vector is ä and parallel to a given vector
5.If coordinates of point A are (x1, y1, z1) and direction ratios of the line are a, b, c, then cartesian form of equation of line is:
6.If coordinates of point A are (x1, y1, z1) and direction cosines of the line are l, m, and n, then Cartesian equation of line is:
7.The vector equation of a line which passes through two points whose position vectors are
8.Cartesian equation of a line which passes through two points (x1, y1, z1) and (x2, y2, z2) is
9.The parametric equations of the line are
10.Equation of the x-axis:
11.Equation of the y-axis:
12.Equation of the z-axis:
13.Conversion of a Cartesian form of an equation of a line to a vector form:
Let the Cartesian form of an equation of line be
Hence, the vector form of the equation of the line is
14.Conversion of a vector form of the equation of a line to the Cartesian form:
Let the Cartesian form of the equation of a line be
and λ is a parameter
Then the Cartesian form of the equation of the line is
15.Angle θ between two lines L1 and L2 passing through the origin and having direction ratios a1, b1, C1 and a2 ,b2, C2 is
16.Condition of perpendicularity: If the lines are perpendicular to each other, then a1a2 + b1b2 + c1c2 = 0.
17.Condition of parallelism: If the lines are parallel, then
18.Equation of a line passing through a point having position vector →k and perpendicular to the lines
19.To find the intersection of two lines:
Consider the two lines:
Step (i): The general coordinates of general points on the given two lines are
Step (iii): Solve the first two equations to get the values of k and m. Check whether the point satisfies the third equation also. If it satisfies, then the lines intersect, otherwise they do not.
Step (iv): Substitute the values of k and m in the set of three equations to get the intersection point.
20.To find the intersection of two lines in the vector form:
Let the two lines be
Step (i): Position vectors of arbitrary points on (1) and (2) are
Step (ii): Because the lines (1) and (2) intersect, they intersect each other, and their points of intersection are as follows:
Step (iii): Solve any two of the equations to get the values of k and m. Substitute the values of k and m in the third equation to check whether it satisfies it. If it does satisfy it, then the two lines intersect, otherwise they do not.
Step (iv): Substitute the values of k and m to get the point of intersection.
21.Perpendicular distance of a line from a point: Let P(u, v, w) be the given point.
Let be the given line.
Let N be the foot of the perpendicular.
Then the coordinates of N are,
Now, the distance PN can be determined using the distance formula.
22. Perpendicular distance of a line from a point when it is in the vector form:
Step (i): Let P(→u) be the given point. Let →r – →a + k→b be the position vector of the line.
Step (ii): Find −−→PN – Position vector of N — Position vector of P
Step (iii): −−→PN.→b = 0
Step (iv): Get the value of k
Step (v): Substitute the value of k in →r – →a + k→b
Step (vi): Compute |−−→PN| to obtain the perpendicular distance.
23.Skew lines: Two lines are said to be skew lines if they are neither parallel nor intersecting.
24.Shortest distance: The shortest distance between two lines L1 and L2 is the distance PQ between the points P and Q, where the lines of shortest distance intersect the two given lines.
25.The shortest distance between two skew lines L and M having equations
26.Condition for two given lines to intersect: If the lines →r = →a1 + k→b1 and →r = →a2 + k→b2 intersect, then the shortest distance between them is zero.
27.The shortest distance between the lines in the Cartesian form
is given by
28.Distance between parallel lines
29.The equation of a plane at a distance d from the origin where ñ is the unit vector normal to the plane, through the origin in vector form, is
30.Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane are I, m, n is lx + my + nz = d.
31.The general equation of the plane is ax + by + cz + d = 0.
32.The equation of a plane perpendicular to a given vector →N and passing through a given point →a is
33.The equation of a plane perpendicular to a given line with direction ratios A, B and C and passing through a given point (x1, y1, z1) is A(x – x1) + B(y – y1) + C(z – z1) = 0
34.The equation of a plane passing through three non-collinear points in the vector form is given as
35.Reduction of the vector form of the equation of a plane to the Cartesian equation:
Let
Then the Cartesian equation of a plane is,
36.The vector equation of the plane passing through the points having position vectors →a, →b and →c is
37.The equation of a plane passing through three non-collinear points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) in the Cartesian form is
38.The intercept form of the equation of a plane is where a, b and c are the intercepts on the x, y and z-axes, respectively.
39.Any plane passing through the intersection of two planes is given by
40.The Cartesian equation of a plane passing through the intersection of two planes
41.The equation of the planes bisecting the angles between the planes
42.The angle θ between a line and the plane ax + by + cz + d = 0 is given by the following relation:
43.If a line is perpendicular to a normal to the plane, then it is parallel to the plane.
44.If a line is parallel to a normal to the plane, then it is perpendicular to the plane.
45.The line →r – →a + k→b lies in the plane →r .→n = d if →a . →n = d and →b . →n = 0
46.The line lies in the plane ax + by + cz + d = 0 if
47.The equation of a plane containing the line
48.The given lines are coplanar if and only
49.Let (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points M and N, respectively. Let a1, b1, c1 and a2, b2, c2 be the direction ratios of →b1 and →b2 , respectively. The given lines are coplanar if and only if,
51.The equation of the plane containing the lines
52.If →n1and →n2 are normals to the planes →r.→n1 = d1 and →r.→n2 = d2, and θ is the angle between the normals drawn from some common point, then
53.Let θ be the angle between two planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0.
The direction ratios of the normal to the planes are A1, B1, C1 and A2, B2, C2.
54.The angle θ between the line and the normal to the plane is given by
55.The distance of point P with position vector →a from a plane where →N is the normal to the plane.
56.The length of the perpendicular from the origin O to the planewhere →N is the normal to the plane.
Important Questions
.
.
Multiple Choice questions-
1. Distance between two planes:
2x + 3y + 4z = 5 and 4x + 6y + 8z = 12 is
(a) 2 units
(b) 4 units
(c) 8 units
(d) 1√29.. units.
2. The planes 2x – y + 4z = 3 and 5x – 2.5y +10 z = 6 are
(a) perpendicular
(b) parallel
(c) intersect along y-axis
(d) passes through (0,0,54..)
3. The co-ordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by:
(a) (2, 0, 0)
(b) (0, 5, 0)
(c) (0, 0, 7)
(d) (0, 5, 7).
4. If α, ß, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction-cosines of the line are:
(a) < sin α, sin ß, sin γ >
(b) < cos α, cos ß, cos γ >
(c) < tan α, tan ß, tan γ >
(d) < cos² α, cos² ß, cos² γ >.
5. The distance of a point P(a, b, c) from x-axis is
(a) √a2+c2
(b) √a2+b2
(c) √b2+c2
(d) b² + c².
6. If the direction-cosines of a line are < k, k, k >, then
(a) k > 0
(b) 0 < k < 1
(c) k = 1
(d) k = 1√3.. or – 1√3..
7. The reflection of the point (α, ß, γ) in the xy-plane is:
(a) (α, ß, 0)
(b) (0, 0, γ)
(c) (-α, -ß, γ)
(d) (α, ß, -γ).
8. What is the distance (in units) between two planes:
3x + 5y + 7z = 3 and 9x + 15y + 21z = 9?
(a) 0
(b) 3
(c) 6√83..
(d) 6.
9. The equation of the line in vector form passing through the point (-1, 3, 5) and parallel to line x−32.. = y−43.., z = 2 is
(a) →r = (-ˆi + 3ˆj + 5ˆk) + λ(2ˆi +3ˆj + ˆk)
(b)→r = (-ˆi+ 3ˆj + 5ˆk) + λ(2ˆi + 3ˆj)
(c) →r = (2ˆi+ 3ˆj– 2ˆk) + λ(-ˆi + 3ˆj + 5ˆk)
(d) →r = (2ˆi + 3ˆj) + λ(-ˆi + 3ˆj + 5ˆk).
10. Let the line x−23.. = y−1−5.. = z−22.. lie in the plane x + 3y – αz + ß = 0. Then (α, ß) equals:
(a) (-6, -17)
(b) (5, -15)ss
(c) (-5, 5)
(d) (6, -17).
Very Short Questions:
1.Find the acute angle which the line with direction-cosines <1√3..,1√6..,n> makes with positive direction of z-axis. (C.B.S.E. Sample Paper 2018-19)
2.Find the direction-cosines of the line.
3.If α, β, γ are direction-angles of a line, prove that cos 2a + cos 2P + cos 2y +1 = 0. (N.C.E.R.T.)
4.Find the length of the intercept, cut off by the plane 2x + y – z = 5 on the x-axis. (C.B.S.E. Outside Delhi 2019)
5.Find the length of the perpendicular drawn from the point P(3, -4,5) on the z-axis.
6.Find the vector equation of a plane, which is at a distance of 5 units from the origin and whosenormal vector is 2ˆi−ˆj+2ˆk
7.If a line makes angles 90°, 135°, 45° with the x,y and z-axes respectively, find its direction cosines.
8.Find the co-ordinates of the point where the line through the points A (3,4,1) and B (5,1, 6) crosses the xy-plane.
9.find the vector equation ofthe line which passes through the point (3,4,5) and is parallel tothe vector 2ˆi + 2ˆj – 3ˆk
Short Questions:
1.Find the acute angle between the lines whose direction-ratios are:
< 1,1,2 > and <-3, -4,1 >.
2.Find the angle between the following pair of lines:
and
and check whether the lines are parallel or perpendicular. (C.B.S.E. 2011)
3.Find the vector equation of the line joining (1.2.3) and (-3,4,3) and show that it is perpendicular to the z-axis. (C.B.S.E. Sample Paper 2018-19)
4.Find the vector equation of the plane, which is 6√29.. at a distance of
units from the origin and its normal vector from the origin is 2ˆi−3ˆj+4ˆk. Also, find its cartesian form. (N.C.E.R.T.)
5.Find the direction-cosines of the unit vector perpendicular to the plane →r⋅(6ˆi−3ˆj−2ˆk) +1 = 0 through the origin. (N.C.E.R.T.)
6.Find the acute angle between the lines
7.Find the angle between the line:
→r=(ˆi−ˆj+ˆk)+λ(2ˆi−ˆj+3ˆk) and the plane →r⋅(2ˆi+ˆj−ˆk)=4 Also, find whether the line is parallel to the plane or not .
8.Find the value of ‘λ’, so that the lines:
1−x3..=7y−14λ..=z−32.. and 7−7x3λ..=y−51..=6−z5.. are at right angles. Also, find whether the lines are intersecting or not
Long Questions:
1.Find the shortest distance between the lines:→r=(4ˆi−ˆj)+λ(ˆi+2ˆj−3ˆk) and →r =(ˆi−ˆj+2ˆk)+μ(2ˆi+4ˆj−5ˆk) (C.B.S.E. 2018)
2.A line makes angles α, β, γ, δ with the four diagonals of a cube, prove that:
3.Find the equation of the plane through the line x−13..=y−42..=z−4−2.. and parallel to the line:
x+12..=1−y4..=z+21..
Hence, find the shortest distance between the lines. (C.B.S.E. Sample Paper 2018-19)
4.Find the Vector and Cartesian equations of the plane passing through the points (2, 2, -1), (3,4,2) and (7,0,6). Also, find the vector equation of a plane passing through (4,3,1) and parallel to the plane obtained above. (C.B.S.E. 2019)
Case Study Questions:
1.Suppose the floor of a hotel is made up of mirror polished Kota stone. Also, there is a large crystal chandelier attached at the ceiling of the hotel. Consider the floor of the hotel as a plane having equation x – 2y + 2z = 3 and crystal chandelier at the point (3, -2, 1).
Based on the above information, answer the following questions.
(i)The d.r’.s of the perpendicular from the point (3, -2, 1) to the plane x – 2y + 2z = 3, is:
a.< 1, 2, 2 >
b.< 1, -2, 2 >
c.< 2, 1, 2 >
d.< 2, -1, 2 >
(ii)The length of the perpendicular from the point (3, -2, 1) to the plane x – 2y + 2z = 3, is:
(iii)The equation of the perpendicular from the point (3, -2, 1) to the plane x – 2y + 2z = 3, is:
(iv)The equation of plane parallel to the plane x – 2y + 2z = 3, which is at a unit distance from the point (3, -2, 1) is:
a.x – 2y + 2z = 0
b.x – 2y + 2z = 6
c.x – 2y + 2z = 12
d.Both (b) and (c)
(v)The image of the point (3, -2, 1) in the given plane is:
2.Consider the following diagram, where the forces in the cable are given.
Based on the above information, answer the following questions.
are 
be the given line.
where a, b and c are the intercepts on the x, y and z-axes, respectively.
is given by 
and the plane ax + by + cz + d = 0 is given by the following relation:
lies in the plane ax + by + cz + d = 0 if
are coplanar if and only 
where →N is the normal to the plane.
where →N is the normal to the plane.
