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). y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1. 6. The shortest distance parallel lines r = a1 + λb1 and r = a2 + μb2 is given by. (ii) The coordinates of any point, which divides the join of points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n internally are (mx2 + nx1 / m + n, my2 + ny1 / m + n, mz2 + nz1 / m + n) ∴ NP = √r2 – p2. Shortest Distance If l1 and l2 are two skew lines, then a line perpendicular to each of lines 4 8. (iv) The equation of a sphere on the line joining two points (x1, y1, z1) and (x2, y2, z2) as a 1 + b2 1 + c2 (a) Lines are parallel, if l1 / 12 = m1 / m2 = n1 / n2 where, 1, m, n are direction cosines of the line. 1 √a2 If the line of shortest distance intersects the lines l1 and l2 at P and Q respectively, then the distance PQ between points P and Q is known as the shortest distance between l1 and l2. 2, ± b2 / √Σ a2 2ux + 2vy + 2wz + d = 0 represents a sphere, if Lines r = a1 + λb1 and r = a2 + μb2 are intersecting lines, if (b1 * b2) * (a2 – a1) = 0. For x intercept Put y = 0, z = 0 in the equation of the plane and obtain the value of x. c is |r – c| = a, (i) The general equation of second degree in x, y, z is ax2 + by2 + cz2 + 2hxy + 2kyz + 2lzx + Similarly, we can determine for other intercepts. 2 + c2 from the plane is equal to. from the centre of the sphere to the plane. (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 5. To Register Online Maths Tuitions on Vedantu.com to clear your doubts from our expert teachers and download the Three Dimensional Geometry formula to solve the problems easily to … b is r = a + λ b, where A, is a parameter. and (0, 0, 1), respectively. (B) represents a real sphere, if u2 +v2 + w2 — d > 0 (vi) The projection of the line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) to the line from the fixed point A. 1, ± b1 / √Σ a2 The coefficient of x, y and z in the cartesian If a, b, c are replaced by direction cosines 1, m, n, then λ, represents Thus, we have x2+y2+z2 + (2u / a) x + (2v / a) y + (2w / a) z + d / a = 0 …..(B) If the plane is given in, normal form lx + my + nz = p. Then, the distance of the point P(x1, y1, (v) The equation of a sphere passing through four non-coplanar points (x1, y1, z1), (x2, y2, z2), 2 + c2 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. (c) Projections of r on the coordinate axes are (vii) The direction ratio of the line passing through points P(x1, y1, z1) and Q(x2, y2, z2) are proportional to x2 – x1, y2 – y1 – z2 – z1 2 + c2 2, Let the plane in the general form be ax + by + cz + d = 0. Then, the Let P(x1, y1, z1) and Q(x2, y2, z2) be two given points. The set of points common to both sphere and plane is l = cos α Math Notes For Class 12 Three Chapter 11: Dimensional Geometry Download PDF. (iii) Since, r2 = u2 + v2 + w2 — d, therefore, the Eq. 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