Class 12 Mathematics Chapter 11 Notes
Class 12 is a very important stage in a student’s academic life. The marks scored in the board exam are considered for competitive exams and admission to prestigious colleges for higher studies. Mathematics is regarded as one of the most challenging subjects. A strong conceptual understanding of all the fundamental concepts and consistent practice are necessary. Using the best reference material, such as the Class 12 Mathematics Chapter 11 notes, helps students to gain a clear understanding of three-dimensional geometry.
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NCERT Class 12 Mathematics Chapter 11 Notes: Key Topics
Introduction:
This Chapter provides a detailed description of 3-D geometry and the coordinate system. Students will learn to use vector algebra, direction cosines and direction ratios of a line joining two points. The Class 12 Mathematics Chapter 11 notes include all equations of planes and lines in Space. The three-dimensional coordinate planes divide Space into a total of eight parts called the Octants.
Representation of a point in Cartesian and Vector form
The point P (1, 2, 3) in the Cartesian form will be represented as (x,y,z)=(1, 2, 3).
In the vector form, the point P is represented as i+2j+3k.
Represent line in 2-D where the endpoints are given as (2,4) and (3,5):-
In the Cartesian form, we write it as
y-4x-3=5-43-2=1
Therefore, (y-3) = 1 (x-3)
In the vector form, we write it as
AB= (b–a) = (3i + 2j) – (5i + 4j)
AB= – 2i – 2j
Represent line in 3-D where the endpoints are (2, 4, 5) and (3, 6, 2)
In the vector form, we write it as
AB= (b–a) = (3i + 6j + 2k) – (2i + 4j + 5k)
AB= – i – 2j +3k
Direction Cosines of a Line
Consider any line passing through the origin (0, 0) and making angles α, β and γ with the x-axis, y-axis and z-axis, respectively; then cosines of the angles, i.e., cosα, cosβ and cosγ are called the direction cosines of the line.
Suppose, cos α =l, cos β =m and cos = n, then l2+ m2+ n2= 1 i.e. cos2α + cos2β+ cos2 =1.
Refer to the Class 12 Mathematics Chapter 11 Notes to get more detailed information on this topic.
Direction Ratios of a Line
Direction ratios are defined as the three numbers which are proportional to the direction cosines of the line. They are denoted by (a, b and c).
For a vector OP= aî +bĵ +ck̂ =r (cos α î + cosβ ĵ + cos γ k̂) where r= a2+b2+c2 then cos α, cos β and cos are the direction cosines, whereas r cos α=a, r cos β =b, r cos =c are termed as direction ratios.
Relationship between the Direction Ratios and Direction Cosines
Suppose a line whose one point is at origin (0, 0) and another point is at P. For a vector OP= aî +bĵ +ck̂ =r (cos α î + cosβ ĵ + cos γ k̂) where r= a2+b2+c2,
we get a=(r cos α) ,b =(r cos β) and c=(r cosγ)
Therefore, we get cos α =(ar), cos β=(br) and cos =(cr).
Hence, cos α = (ar)= (±) aa2+b2+c2
Similarly, cos β=(br)= (±) ba2+b2+c2 and cos = (cr)= (±) ca2+b2+c2.
Relation between the direction ratios and cosines when a line is passing through two points:
Only one line passes through the given two points. For a line PQ, and the coordinates are given as P(x1, y1, z1) and Q(x2, y2, z2). Consider l, m and n as the direction cosines of line PQ that will make an angle α, β and with the x-axis, y-axis and z-axis, respectively.
Hence, cos α= x2–x1PQ , cosβ = y2–y1PQ and cosγ = z2–z1PQ, where PQ=(x2–x1)2+(y2–y1)2+(z2–z1)2
The direction ratios of a line segment can also be considered as (x2–x1, y2–y1, z2–z1 or x1–x2, y1–y2, z1–z2).
Equation of a line in Space
- A line passing through a point and is parallel to a given vector is given as r = a + b, where
a = position vector at a point A w.r.t. the origin and is parallel to vector b,
l = line passing through the point A and vector b and
r = position vector at P
= real number.
In Cartesian form:
x-x1a=y-y1b=z-z1c, where coordinates of point A is (x1, y1, z1) and direction ratios of the given line be a, b, and c. Coordinates of point P be (x, y, z).
Then r = xî +yĵ +zk̂, a = x1î +y1ĵ +z1k̂
2. A line passing through points A(x1, y1, z1) and B(x2, y2, z2) will be r = a + (b–a), where
a and b = position vectors
r = position vector at P(x, y, z)
= real number.
In Cartesian form:
x-x1x2–x1=y-y1y2–y1=z-z1z2–z1, where x = x1 + (x2–x1), y = y1 + (y2–y1) and z = z1 + (z2–z1)
Solve unlimited problems included in the Class 12 Mathematics Chapter 11 Notes at Extramarks website.
Angle between the two lines with respect to their direction ratios:
Consider L1 and L2 as two lines which pass through the origin (0, 0). L1 and L2 has direction ratios a1,b1, c1 and a2, b2, c2, respectively. If P is a point on the line L1 and Q is a point on the line L2, then the directed lines will be OP and OQ. Suppose, q is the acute angle between OP and OQ, then the angle q is given by
Cos = a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22 and Sin = (a1b2–a2b1)2+(b1c2–b2c1)2+(c1a2–c2a1)2a12+b12+c12a22+b22+c22
The direction ratios of the two lines a1,b1, c1 and a2, b2, c2 are
(i) perpendicular i.e. if q = 90° and
(ii) parallel i.e. if q = 0
Distance between two lines:
- For two skew lines, r1 = a1 + b1 and r2 = a2 + b2
Let PQ be the shortest distance between lines l1 and l2. If n is the unit vector along PQ, then
n = b1 X b2b1 X b2. If PQ = d. n, where d is the magnitude of distance.
d= (b1 X b2) . (a1 X a2)b1 X b2
2. For two parallel lines, r1 = a1 + b1 and r2 = a2 + b2
l1 and l2 are coplanar lines as they are parallel, d = PT= b . (a1 – a2)b
PLANE:
A plane can be determined uniquely if:
(i) the normal to the plane and the distance from the origin (0, 0) is given, that is, the equation of the plane in the normal form.
(ii) it passes through any point and is perpendicular to the direction.
(iii) it passes through the given non-collinear points.
Equation in normal form:
If d is the distance from the origin, ON is the normal and n= unit normal. Suppose NP is perpendicular to ON and r is the position vector at P(x, y, z), then r.n = d is the vector equation and lx + my + nz = d is the Cartesian equation of the plane
Refer to the Class 12 Chapter 11 Mathematics notes to access unlimited problems for extra practice.
Equation of a given plane which is perpendicular to a vector and passes through any point:
Consider a plane passing through point A with a position vector a perpendicular to the vector N. If r is the position vector of the point P(x, y, z), then (r – a). N = 0 is the vector equation and A (x-x1) + B (y-y1) + C (z-z1) = 0 is the cartesian equation of the plane.
Equation of a plane through three non-collinear points
Let points R, S and T be non-collinear on the plane having position vectors a, b, and c, respectively.
The equation of the plane through point R and perpendicular to the RS and RT in the vector equation is given as (r – a) [(b – a) x (c – a)] = 0. The Cartesian equation is in the form of determinants, we have |D| = 0, where D =
| x-x1 |
y-y1 |
z-z1 |
| x-x2 |
y-y2 |
z-z2 |
| x-x3 |
y-y3 |
z-z3 |
Intercept form of an equation:
Consider Ax + By + Cz + D = 0 as the equation of the plane. If a, b, c are the intercepts on x, y and z-axes at (a, 0, 0), (0, b, 0) and also (0, 0, c) respectively. Then the required plane equation in the form of intercepts is given as xa+yb+zc=0.
The Class 12 Mathematics Chapter 11 Notes includes several questions based on this concept. Students must consider referring to these notes for extra practice.
Equation of a given plane passing through the intersection of two different planes:
If P1 and P2 are two planes having equations r.n1 = d1 and r.n2 = d2, where r is the position vector, then we express any plane which passes through the intersection of two planes as r.(n1+n2) = d1+ d2 in the vector equation.
In Cartesian equation, we have
(A1x + B1y + C1z – d1) + (A2x + B2y + C2z – d2) = 0
Coplanarity of Two Lines:
Consider lines r = a1 + b1 and r = a2 + b2, if AB is perpendicular to b1 X b2. Then in vector form, we have AB. (b1 X b2)= 0 or (a2–a1).(b1 X b2) = 0. In cartesian form, the equation is expressed as |D|=0, where D =
| x2–x1 |
y2–y1 |
z2–z1 |
| a1 |
b1 |
c1 |
| a2 |
b2 |
c2 |
Angle between Two Planes
Cos = n1.n2n1n2, where is the angle between the two planes drawn from a common point.
In Cartesian form, Cos = a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22 where a1,b1, c1 and a2, b2, c2 are the direction ratios of the equation of planes.
Angle between line and plane:
The angle between a plane r.n1 = d1 and a line r = a + b is given as:
In vector form: Cos = Sin = b . nbn
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Class 12 Mathematics Chapter 11 Notes: Exercise & Solutions
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NCERT Class 12 Mathematics Chapter 11 Notes: Key Features
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