Two planes are parallel or perpendicular according as the normals to them are parallel or perpendicular. (ii) Any sphere concentric with the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 is x2 + y2 + z2 + 2ux + 2vy + 2wz + k = 0 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. 1 and ± a2 / √Σ a2 (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 general equation of the first degree in x, y, z always represents a plane. 7. 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 A plane is a surface such that, if two points are taken on it, a straight line joining them lies wholly in the surface. and 12 is known as the line of shortest distance. 2, ± c2 / √Σ a2 2. The intercept form of equation of plane represented in the form of x / a + y / b + z / c = 1 8. Then, direction cosines of PQ arex2 – x1 / |PQ|, y2 – y1 / |PQ|, z2 – z1 / |PQ|, If the vertices of a triangle be A(x1, y1, z1) and B(x2, y2, z2) and C(x3, y3, z3), then, If l(x1, m1, n1) and l(x2, m2, n2) be the direction cosines of two given lines, then the angle θ between them is given by cos θ = l112 + m1m2 + n1n2, (i) The angle between any two diagonals of a cube is cos-1 (1 / 3). Math Notes For Class 12 Three Chapter 11: Dimensional Geometry Download PDF. If the plane is given in, normal form lx + my + nz = p. Then, the distance of the point P(x1, y1, We place the coordinates in brackets. 2ux + 2vy + 2wz + d = 0 represents a sphere, if = √u2 + v2 + w2 – ad / |a| . (x3, y3, z3) and (x4, y4, z4) is, The plane lx + my + nz = p will touch the sphere x2 + y2 + z2 + 2ux + 2vy + 2 wz + d = 0, if length of the perpendicular from the centre ( – u, – v,— w)= radius, i.e., |lu – mv – nw – p| / √l2 + m2 + n2 where, a, b and c are intercepts on X, Y and Z-axes, respectively. 2, If θ is the angle between the normals, then cos θ = ± a1a2 + b1b2 + c1c2 / √a21 + b2 Chapter 5: Continuity and Differentiability, Chapter 2: Inverse Trigonometric Functions, NCERT Solutions for Class 9 Science Maths Hindi English Math, NCERT Solutions for Class 10 Maths Science English Hindi SST, Class 11 Maths Ncert Solutions Biology Chemistry English Physics, Class 12 Maths Ncert Solutions Chemistry Biology Physics pdf, Class 1 Model Test Papers Download in pdf, Class 5 Model Test Papers Download in pdf, Class 6 Model Test Papers Download in pdf, Class 7 Model Test Papers Download in pdf, Class 8 Model Test Papers Download in pdf, Class 9 Model Test Papers Download in pdf, Class 10 Model Test Papers Download in pdf, Class 11 Model Test Papers Download in pdf, Class 12 Model Test Papers Download in pdf. (iii) Since, r2 = u2 + v2 + w2 — d, therefore, the Eq. Vector equation of a line passing through a point with position vector a and parallel to vector 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. 3). 1 √a2 The y - coordinate is also called the ordinate. 1, ± b1 / √Σ a2 (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 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 where, 1, m, n are direction cosines of the line. 1 + b2 1 + c2 To find its centre and radius first we make the coefficients of x2, y2 and z2 each unity by (iv) Coordinates of the centroid of a triangle formed with vertices P(x1, y1, z1) and Q(x2, y2, z2) Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) CBSE Syllabus Class 12 Maths Physics Chemistry ... CBSE Syllabus Class 11 Mathematics biology chemistry ... CBSE Syllabus Class 10 Maths Science Hindi English ... CBSE Syllabus Class 9 Mathematics Science English Hindi ... Revised Syllabus for Class 12 Mathematics. 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. Hence, the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are parallel, if a1 / a2 = b1 / b2 = c1 / c2 and perpendicular, if a1a2 + b1b2 + c1c2 = 0. and (0, 0, 1), respectively. The equation of the sphere with centre (a, b, c) and radius r is (x – a)2 + (y – b)2 + (z – c)2 = r2 ……. diameter is The equation becomes ax2 + ay2 + az2 + 2ux + 2vy + 2wz + d – 0 …(A). of a great circle are the same as those of the sphere. 1 + b2 Any point P on this line may be taken as (x1 + λa, y1 + λb, z1 + λc), where λ ∈ R is parameter. 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 2 + c2 1. The distance between these points is (b) Lines are perpendicular, if l112 + m1m2 + n1n2 from the centre of the sphere to the plane. plane. 2 + b2 (c) Projections of r on the coordinate axes are z1) from the plane is |lx1 + my1 + nz1 – p|. l = cos α 2, ± b2 / √Σ a2 6. If ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 be equation of two parallel planes. only if, (v) Equation of the plane containing two coplanar lines, The angle between two planes is defined as the angle between the normal to them from any point. (iv) The equation of a sphere on the line joining two points (x1, y1, z1) and (x2, y2, z2) as a and radius = √u2 / a2 + v2 / a2 + w2 / a2 – d / a from the plane is equal to. Free PDF download of Three Dimensional Geometry Formulas for CBSE Class 12 Maths. Hence, the locus of P is a circle whose centre is at the point N, the foot of the perpendicular 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. Then, we say that the direction ratio of r are proportional to a, b, Also, we have l = a / √a2 + b2 + c2, m = b / √a2 + b2 + c2, n = c / √a2 + b2 + c2 i.e., m = cos β and n = cos γ, If OP = r, then coordinates of OP are (lr, mr , nr), (i) If 1, m, n are direction cosines of a vector r, then (v) If θ is the angle between two lines whose direction ratios are proportional to a1, b1, c1 and (i) 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 (b) h = k = 1 = 0 The coefficient of x, y and z in the cartesian 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. Let P(x1, y1, z1) and Q(x2, y2, z2) be two given points. If (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) are the vertices of a tetrahedron, then its centroid G is given by (vi) The projection of the line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) to the line with direction ratios a, b, c is, (iii) The Equation of a plane passing through three points A(x1, y1, z1), B(x2, y2, z2) and C(x3, nor intersecting. from the fixed point A. (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 Lines r = a1 + λb1 and r = a2 + μb2 are intersecting lines, if (b1 * b2) * (a2 – a1) = 0. 2 + c2 dividing throughout by a. Copyright @ ncerthelp.com A free educational website for CBSE, ICSE and UP board. 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. 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