Three Dimensional Geometry Coordinate System The three mutually perpendicular lines in a space which divides the space into eight parts and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system. equation of the plane is ax + by + cz + d = 0. We place the coordinates in brackets. A plane is a surface such that, if two points are taken on it, a straight line joining them lies wholly in the surface. (iv) The equation of a sphere on the line joining two points (x1, y1, z1) and (x2, y2, z2) as a A sphere is the locus of a point which moves in a space in such a way that its distance from a fixed point always remains constant. (ii) The angle between a diagonal of a cube and the diagonal of a face (of the cube is cos-1 (√2 / (b) Lines are perpendicular, if l112 + m1m2 + n1n2 having direction cosines 1, m, n is |(x2 – x1)l + (y2 – y1)m + (z2 – z1)n|. 1 + b2 1 + c2 2, Let the plane in the general form be ax + by + cz + d = 0. (iv) If θ is the angle between two lines having direction cosines l1, m1, n1 and 12, m2, n2, then cos θ = l112 + m1m2 + n1n2 (ii) The coordinates of foot of perpendicular N from the origin on the plane are (1p, mp, np). y1) + c (z — z1) = 0. In generally, we can write x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 Shortest Distance If l1 and l2 are two skew lines, then a line perpendicular to each of lines 4 The y - coordinate is also called the ordinate. 1, ± c1 / √Σ a2 l = cos α 2 + c2 where, 1, m, n are direction cosines of the line. (iii) Since, r2 = u2 + v2 + w2 — d, therefore, the Eq. The foot (x, y, z) of a point (x1, y1, z1) in a plane ax + by + cz + d = 0 is given by x – x1 / a = y – y1 / b = z – z1 / c = – (ax1 + by1 + cz1 + d) / a2 + b2 + c2, 14. Copyright @ ncerthelp.com A free educational website for CBSE, ICSE and UP board. and R(x3, y3, z3) are (x1 + x2 + x3 / 3 , y1 + y2 + y3 / 3, z1 + z2 + z3 / 3) y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1. cz + k = 0, where k may be determined from given conditions. Hence, the general 1 √a22 + b2 Lines are perpendicular, if a1a2 + b1b2 + c1c2 = 0. Vector equation of a line passing through a point with position vector a and parallel to vector Therefore, their equations are 1 + c2 (vi) The projection of the line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) to the line in the ratio m : n internally are (mx2 + nx1 / m + n, my2 + ny1 / m + n, mz2 + nz1 / m + n) 2. c is |r – c| = a, (i) The general equation of second degree in x, y, z is ax2 + by2 + cz2 + 2hxy + 2kyz + 2lzx + 2ux + 2vy + 2wz + d = 0 represents a sphere, if Thus, the angle between the two planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0, is equal to the angle between the normals with direction cosines ± a1 / √Σ a2 8. 2 + c2 nor intersecting. + 3 and for Q, it is –2. NCERT Notes for Class 12 Mathematics Chapter 11: Three Dimensional Geometry Coordinate System. 2 + b2 from the fixed point A. Lines are parallel, if a1 / a2 = b1 / b2 = c1 / c2 Here, its centre is (-u, v, w) and radius = √u2 + v2 + w2 – d, The vector equation of a sphere of radius a and Centre having position vector In Vector Form The angle between a line r = a + λ b and plane r *• n = d, is defined as the complement of the angle between the line and normal to the plane: In Cartesian Form The angle between a line x – x1 / a1 = y – y1 / b1 = z – z1 / c1 and plane a2x + b2y + c2z + d2 = 0 is sin θ = a1a2 + b1b2 + c1c / √a2 The set of points common to both sphere and plane is Please send your queries to ncerthelp@gmail.com you can aslo visit our facebook page to get quick help. Mathematics Notes for Class 12 chapter 11. r = a + λ (b – a) , where λ is a parameter. Hence, the locus of P is a circle whose centre is at the point N, the foot of the perpendicular distance of the point P b is r = a + λ b, where A, is a parameter. The shortest distance between lines r = a1 + λb1 and r = a2 + μb2 is given by, 10. 2 + b2 1 + b2 Note The equation of plane parallel to a given plane ax + by + cz + d = 0 is given by ax + by + (i) Equation of a plane passing through the point A(x1, y1, z1) and parallel to two given lines with direction ratios, (ii) Equation of a plane through two points A(x1, y1, z1) and B(x2, y2, z2) and parallel to a line Link of our facebook page is given in sidebar. Then, the Any point P on this line may be taken as (x1 + λa, y1 + λb, z1 + λc), where λ ∈ R is parameter. m, n. Then, x2 – x1 = l|PQ|, y2 – y1 = m|PQ|, z2 – z1 = n|PQ| These are projections of PQ on X , Y and Z axes, respectively. 2. Vector equation of a line passing through two given points having position vectors a and b is = √u2 + v2 + w2 – ad / |a| . (ii) Any sphere concentric with the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 is x2 + y2 + z2 + 2ux + 2vy + 2wz + k = 0 (v) If θ is the angle between two lines whose direction ratios are proportional to a1, b1, c1 and 3). Skew Lines Two straight lines in space are said to be skew lines, if they are neither parallel The image or reflection (x, y, z) of a point (x1, y1, z1) in a plane ax + by + cz + d = 0 is given by x – x1 / a = y – y1 / b = z – z1 / c = – 2 (ax1 + by1 + cz1 + d) / a2 + b2 + c2, 13. (a) r = |r| (li + mj + nk) ⇒ r = li + mj + nk One of these planes will bisect the acute angle and the other obtuse angle between the given i.e., Similarly, we can determine for other intercepts. ∴ Centre is (- u / a, – v / a, – w / a) = √u2 + v2 + w2 – d (lu – mv – nw – p)2 = (u2 + v2 + w2 – d) (l2 + m2 + n2), Consider a sphere intersected by a plane. (x1 + x2 + x3 + x4 / 4 , y1 + y2 + y3 + y4 / 4, z1 + z2 + z3 + z4 / 4), If a directed line segment OP makes angle α, β and γ with OX , OY and OZ respectively, then 2, ± b2 / √Σ a2 1 √a2 The two equations of the line ax + by + cz + d = 0 and a’ x + b’ y + c’ z + d’ = 0 together represents a straight line. (a) a = b = c (≠ 0) y3, z3) is, (iv) Four points A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) are coplanar if and (iii) In stating the coordinates of a point in the coordinate plane, the x - coordinate comes first, and then the y - coordinate. Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) 6. a2, b2, c2 respectively, then the angle θ between them is given by cos θ = a1a2 + b1b2 + c1c2 / √a2 11. For x intercept Put y = 0, z = 0 in the equation of the plane and obtain the value of x. (c) Projections of r on the coordinate axes are and (0, 0, 1), respectively. plane. The section of sphere by a plane through its centre is called a great circle. 1 = ± a2x + b2y + c2z + d2 / √Σa2 The centre and radius Equation of a straight line joining two fixed points A(x1, y1, z1) and B(x2, y2, z2) is given by x – x1 / x2 – x1 = y – y1 / y2 – y1 = z – z1 / z2 – z1. (a) Lines are parallel, if l1 / 12 = m1 / m2 = n1 / n2 (d) |r| = l|r|, m|r|, n|r| / √sum of the squares of projections of r on the coordinate axes, (ii) If P(x1, y1, z1) and Q(x2, y2, z2) are two points, such that the direction cosines of PQ are l, 1, ± b1 / √Σ a2 equation of a plane are the direction ratios of normal to the plane. 1. The equation becomes ax2 + ay2 + az2 + 2ux + 2vy + 2wz + d – 0 …(A). Two planes are parallel or perpendicular according as the normals to them are parallel or perpendicular. (a) The length of the perpendicular from a point on the line r – a + λ b is given by, (b) The length of the perpendicular from a point P(x1, y1, z1) on the line. The coefficient of x, y and z in the cartesian 2, ± c2 / √Σ a2 2 Equation of a straight line passing through a fixed point A(x1, y1, z1) and having direction ratios a, b, c is given by x – x1 / a = y – y1 / b = z – z1 / c, it is also called the symmetrically form of a line.