{"id":28286,"date":"2022-01-13T18:14:05","date_gmt":"2022-01-13T12:44:05","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=28286"},"modified":"2022-04-25T15:18:42","modified_gmt":"2022-04-25T09:48:42","slug":"three-dimensional-geometry-class-12-notes-maths-chapter-11","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-materials\/three-dimensional-geometry\/class-12-notes\/maths-chapter-11\/","title":{"rendered":"Three Dimensional Geometry Class 12 Notes Maths Chapter 11"},"content":{"rendered":"

Direction Cosines of a Line:<\/strong> If the directed line OP makes angles \u03b1, \u03b2, and \u03b3 with positive X-axis, Y-axis and Z-axis respectively, then cos \u03b1, cos \u03b2, and cos \u03b3, are called direction cosines of a line. They are denoted by l, m, and n. Therefore, l = cos \u03b1, m = cos \u03b2 and n = cos \u03b3. Also, sum of squares of direction cosines of a line is always 1,
\ni.e. l2<\/sup> + m2<\/sup> + n2<\/sup> = 1 or cos2<\/sup> \u03b1 + cos2<\/sup> \u03b2 + cos 2<\/sup> \u03b3 = 1
\nNote: Direction cosines of a directed line are unique.<\/p>\n

Direction Ratios of a Line:<\/strong> Number proportional to the direction cosines of a line, are called direction ratios of a line.
\n(i) If a, b and c are direction ratios of a line, then \\(\\frac { l }{ a }\\) = \\(\\frac { m }{ b }\\) = \\(\\frac { n }{ c }\\)
\n(ii) If a, b and care direction ratios of a line, then its direction cosines are
\n\"Three
\n(iii) Direction ratios of a line PQ passing through the points P(x1<\/sub>, y1<\/sub>, z1<\/sub>) and Q(x2<\/sub>, y2<\/sub>, z2<\/sub>) are x2<\/sub> \u2013 x1<\/sub>, y2<\/sub> \u2013 y1<\/sub> and z2<\/sub> \u2013 z1<\/sub> and direction cosines are
\n\"Three
\n\"Three
\nNote:
\n(i) Direction ratios of two parallel lines are proportional.
\n(ii) Direction ratios of a line are not unique.<\/p>\n

Straight line:<\/strong> A straight line is a curve, such that all the points on the line segment joining any two points of it lies on it.<\/p>\n

Equation of a Line through a Given Point and parallel to a given vector \\(\\vec { b }\\)
\nVector form \\(\\vec { r } =\\vec { a } +\\lambda \\vec { b }\\)
\nwhere, \\(\\vec { a }\\) = Position vector of a point through which the line is passing
\n\\(\\vec { b }\\) = A vector parallel to a given line<\/p>\n

Cartesian form
\n\"Three
\nwhere, (x1<\/sub>, y1<\/sub>, z1<\/sub>) is the point through which the line is passing through and a, b, c are the direction ratios of the line.
\nIf l, m, and n are the direction cosines of the line, then the equation of the line is
\n\"Three
\nRemember point: Before we use the DR\u2019s of a line, first we have to ensure that coefficients of x, y and z are unity with a positive sign.<\/p>\n

Equation of Line Passing through Two Given Points<\/strong>
\nVector form:<\/strong> \\(\\vec { r } =\\vec { a } +\\lambda \\left( \\vec { b } -\\vec { a } \\right)\\), \u03bb \u2208 R, where a and b are the position vectors of the points through which the line is passing.<\/p>\n

Cartesian form<\/strong>
\n\"Three
\nwhere, (x1<\/sub>, y1<\/sub>, z1<\/sub>) and (x2<\/sub>, y2<\/sub>, z2<\/sub>) are the points through which the line is passing.<\/p>\n

Angle between Two Lines<\/strong>
\nVector form:<\/strong> Angle between the lines \\(\\vec { r } =\\vec { { a }_{ 1 } } +\\lambda \\vec { { b }_{ 1 } }\\) and \\(\\vec { r } =\\vec { { a }_{ 2 } } +\\mu \\vec { { b }_{ 2 } }\\) is given as
\n\"Three
\n\"Three<\/p>\n

Condition of Perpendicularity:<\/strong> Two lines are said to be perpendicular, when in vector form \\(\\vec { { b }_{ 1 } } \\cdot \\vec { { b }_{ 2 } } =0\\); in cartesian form a1<\/sub>a2<\/sub> + b1<\/sub>b2<\/sub> + c1<\/sub>c2<\/sub> = 0
\nor l1<\/sub>l2<\/sub> + m1<\/sub>m2<\/sub> + n1<\/sub>n2<\/sub> = 0 [direction cosine form]\n

Condition that Two Lines are Parallel:<\/strong> Two lines are parallel, when in vector form \\(\\vec { { b }_{ 1 } } \\cdot \\vec { { b }_{ 2 } } =\\left| \\vec { { b }_{ 1 } } \\right| \\left| \\vec { { b }_{ 2 } } \\right|\\); in cartesian form \\(\\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\\frac { { c }_{ 1 } }{ { c }_{ 2 } }\\)
\nor
\n\\(\\frac { { l }_{ 1 } }{ { l }_{ 2 } } =\\frac { { m }_{ 1 } }{ { m }_{ 2 } } =\\frac { { n }_{ 1 } }{ { n }_{ 2 } }\\)
\n[direction cosine form]\n

Shortest Distance between Two Lines:<\/strong> Two non-parallel and non-intersecting straight lines, are called skew lines.
\nFor skew lines, the line of the shortest distance will be perpendicular to both the lines.
\nVector form:<\/strong> If the lines are \\(\\vec { r } =\\vec { { a }_{ 1 } } +\\lambda \\vec { { b }_{ 1 } }\\) and \\(\\vec { r } =\\vec { { a }_{ 2 } } +\\lambda \\vec { { b }_{ 2 } }\\). Then, shortest distance
\n\"Three
\nwhere \\(\\vec { { a }_{ 2 } }\\), \\(\\vec { { a }_{ 1 } }\\) are position vectors of point through which the line is passing and \\(\\vec { { b }_{ 1 } }\\), \\(\\vec { { b }_{ 2 } }\\) are the vectors in the direction of a line.<\/p>\n

Cartesian form:<\/strong> If the lines are
\n\"Three
\nThen, shortest distance,
\n\"Three<\/p>\n

Distance between two Parallel Lines:<\/strong> If two lines l1<\/sub> and l2<\/sub> are parallel, then they are coplanar. Let the lines be \\(\\vec { r } =\\vec { { a }_{ 1 } } +\\lambda \\vec { b }\\) and \\(\\vec { r } =\\vec { { a }_{ 2 } } +\\mu \\vec { b }\\), then the distance between parallel lines is
\n\"Three
\nNote: If two lines are parallel, then they both have same DR\u2019s.<\/p>\n

Distance between Two Points:<\/strong> The distance between two points P (x1<\/sub>, y1<\/sub>, z1<\/sub>) and Q (x2<\/sub>, y2<\/sub>, z2<\/sub>) is given by
\n\"Three<\/p>\n

Mid-point of a Line:<\/strong> The mid-point of a line joining points A (x1<\/sub>, y1<\/sub>, z1<\/sub>) and B (x2<\/sub>, y2<\/sub>, z2<\/sub>) is given by
\n\"Three<\/p>\n

Plane:<\/strong> A plane is a surface such that a line segment joining any two points of it lies wholly on it. A straight line which is perpendicular to every line lying on a plane is called a normal to the plane.<\/p>\n

Equations of a Plane in Normal form<\/strong>
\nVector form:<\/strong> The equation of plane in normal form is given by \\(\\vec { r } \\cdot \\vec { n } =d\\), where \\(\\vec { n }\\) is a vector which is normal to the plane.
\nCartesian form:<\/strong> The equation of the plane is given by ax + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from origin.
\nAnother equation of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from origin and p is a distance of a plane from origin.
\nNote: If d is the distance from the origin and l, m and n are the direction cosines of the normal to the plane through the origin, then the foot of the perpendicular is (ld, md, nd).<\/p>\n

Equation of a Plane Perpendicular to a given Vector and Passing Through a given Point<\/strong>
\nVector form:<\/strong> Let a plane passes through a point A with position vector \\(\\vec { a }\\) and perpendicular to the vector \\(\\vec { n }\\), then \\(\\left( \\vec { r } -\\vec { a } \\right) \\cdot \\vec { n } =0\\)
\nThis is the vector equation of the plane.
\nCartesian form:<\/strong> Equation of plane passing through point (x1<\/sub>, y1<\/sub>, z1<\/sub>) is given by
\na (x \u2013 x1<\/sub>) + b (y \u2013 y1<\/sub>) + c (z \u2013 z1<\/sub>) = 0 where, a, b and c are the direction ratios of normal to the plane.<\/p>\n

Equation of Plane Passing through Three Non-collinear Points<\/strong>
\nVector form:<\/strong> If \\(\\vec { a }\\), \\(\\vec { b }\\) and \\(\\vec { c }\\) are the position vectors of three given points, then equation of a plane passing through three non-collinear points is \\(\\left( \\vec { r } -\\vec { a } \\right) \\cdot \\left\\{ \\left( \\vec { b } -\\vec { a } \\right) \\times \\left( \\vec { c } -\\vec { a } \\right) \\right\\} =0\\).
\nCartesian form:<\/strong> If (x1<\/sub>, y1<\/sub>, z1<\/sub>) (x2<\/sub>, y2<\/sub>, z2<\/sub>) and (x3<\/sub>, y3<\/sub>, z3<\/sub>) are three non-collinear points, then equation of the plane is
\n\"Three
\nIf above points are collinear, then
\n\"Three<\/p>\n

Equation of Plane in Intercept Form:<\/strong> If a, b and c are x-intercept, y-intercept and z-intercept, respectively made by the plane on the coordinate axes, then equation of plane is \\(\\frac { x }{ a } +\\frac { y }{ b } +\\frac { z }{ c } =1\\)<\/p>\n

Equation of Plane Passing through the Line of Intersection of two given Planes<\/strong>
\nVector form:<\/strong> If equation of the planes are \\(\\vec { r } \\cdot \\vec { { n }_{ 1 } } ={ d }_{ 1 }\\) and \\(\\vec { r } \\cdot \\vec { { n }_{ 2 } } ={ d }_{ 2 }\\), then equation of any plane passing through the intersection of planes is
\n\\(\\vec { r } \\cdot \\left( \\vec { { n }_{ 1 } } +\\lambda \\vec { { n }_{ 2 } } \\right) ={ d }_{ 1 }+\\lambda { d }_{ 2 }\\)
\nwhere, \u03bb is a constant and calculated from given condition.
\nCartesian form:<\/strong> If the equation of planes are a1<\/sub>x + b1<\/sub>y + c1<\/sub>z = d1<\/sub> and a2<\/sub>x + b2<\/sub>y + c2<\/sub>z = d2<\/sub>, then equation of any plane passing through the intersection of planes is a1<\/sub>x + b1<\/sub>y + c1<\/sub>z \u2013 d1<\/sub> + \u03bb (a2<\/sub>x + b2<\/sub>y + c2<\/sub>z \u2013 d2<\/sub>) = 0
\nwhere, \u03bb is a constant and calculated from given condition.<\/p>\n

Coplanarity of Two Lines<\/strong>
\nVector form:<\/strong> If two lines \\(\\vec { r } =\\vec { { a }_{ 1 } } +\\lambda \\vec { { b }_{ 1 } }\\) and \\(\\vec { r } =\\vec { { a }_{ 2 } } +\\mu \\vec { { b }_{ 2 } }\\) are coplanar, then
\n\\(\\left( \\vec { { a }_{ 2 } } -\\vec { { a }_{ 1 } } \\right) \\cdot \\left( \\vec { { b }_{ 2 } } -\\vec { { b }_{ 1 } } \\right) =0\\)
\n\"Three<\/p>\n

Angle between Two Planes: Let \u03b8 be the angle between two planes<\/a>.<\/strong>
\nVector form:<\/strong> If \\(\\vec { { n }_{ 1 } }\\) and \\(\\vec { { n }_{ 2 } }\\) are normals to the planes and \u03b8 be the angle between the planes \\(\\vec { r } \\cdot \\vec { { n }_{ 1 } } ={ d }_{ 1 }\\) and \\(\\vec { r } \\cdot \\vec { { n }_{ 2 } } ={ d }_{ 2 }\\), then \u03b8 is the angle between the normals to the planes drawn from some common points.
\n\"Three
\nNote: The planes are perpendicular to each other, if \\(\\vec { { n }_{ 1 } } \\cdot \\vec { { n }_{ 2 } } =0\\) and parallel, if \\(\\vec { { n }_{ 1 } } \\cdot \\vec { { n }_{ 2 } } =\\left| \\vec { { n }_{ 1 } } \\right| \\left| \\vec { { n }_{ 2 } } \\right|\\)
\nCartesian form:<\/strong> If the two planes are a1<\/sub>x + b1<\/sub>y + c1<\/sub>z = d1<\/sub> and a2<\/sub>x + b2<\/sub>y + c2<\/sub>z = d2<\/sub>, then
\n\"Three
\nNote: Planes are perpendicular to each other, if a1<\/sub>a2<\/sub> + b1<\/sub>b2<\/sub> + c1<\/sub>c2<\/sub> = 0 and planes are parallel, if \\(\\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\\frac { { c }_{ 1 } }{ { c }_{ 2 } }\\)<\/p>\n

Distance of a Point from a Plane<\/strong>
\nVector form:<\/strong> The distance of a point whose position vector is \\(\\vec { a }\\) from the plane
\n\\(\\vec { r } \\cdot \\hat { n } =d\\quad is\\quad \\left| d-\\vec { a } \\hat { n } \\right|\\)<\/p>\n

Note:
\n(i) If the equation of the plane is in the form \\(\\vec { r } \\cdot \\vec { n } =d\\), where \\(\\vec { n }\\) is normal to the plane, then the perpendicular distance is \\(\\frac { \\left| \\vec { a } \\cdot \\vec { n } -d \\right| }{ \\left| \\vec { n } \\right| }\\)
\n(ii) The length of the perpendicular from origin O to the plane \\(\\vec { r } \\cdot \\vec { n } =d\\quad is\\quad \\frac { \\left| d \\right| }{ \\left| \\vec { n } \\right| }\\) [\u2235 \\(\\vec { a }\\) = 0]\n

Cartesian form:<\/strong> The distance of the point (x1<\/sub>, y1<\/sub>, z1<\/sub>) from the plane Ax + By + Cz = D is
\n\"Three<\/p>\n

Angle between a Line and a Plane<\/strong>
\nVector form:<\/strong> If the equation of line is \\(\\vec { r } =\\vec { a } +\\lambda \\vec { b }\\) and the equation of plane is \\(\\vec { r } \\cdot \\vec { n } =d\\), then the angle \u03b8 between the line and the normal to the plane is
\n\"Three
\nand so the angle \u03a6 between the line and the plane is given by 90\u00b0 \u2013 \u03b8,
\ni.e. sin(90\u00b0 \u2013 \u03b8) = cos \u03b8
\n\"Three<\/p>\n

Cartesian form:<\/strong> If a, b and c are the DR\u2019s of line and lx + my + nz + d = 0 be the equation of plane, then
\n\"Three
\nIf a line is parallel to the plane, then al + bm + cn = 0 and if line is perpendicular to the plane, then \\(\\frac { a }{ l } =\\frac { b }{ m } =\\frac { c }{ n }\\)<\/p>\n

Remember Points<\/strong>
\n(i) If a line is parallel to the plane, then normal to the plane is perpendicular to the line. i.e. a1<\/sub>a2<\/sub> + b1<\/sub>b2<\/sub> + c1<\/sub>c2<\/sub> = 0
\n(ii) If a line is perpendicular to the plane, then DR\u2019s of line are proportional to the normal of the plane.
\ni.e. \\(\\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\\frac { { c }_{ 1 } }{ { c }_{ 2 } }\\)
\nwhere, a1<\/sub>, b1<\/sub> and c1<\/sub> are the DR\u2019s of a line and a2<\/sub>, b2<\/sub> and c2<\/sub> are the DR\u2019s of normal to the plane.<\/p>\n

<\/h5>\n","protected":false},"excerpt":{"rendered":"

Direction Cosines of a Line: If the directed line OP makes angles \u03b1, \u03b2, and \u03b3 with positive X-axis, Y-axis […]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"Three Dimensional Geometry Class 12 Notes Maths Chapter 11","_yoast_wpseo_metadesc":"CBSE Class 12 Maths Notes Chapter 11 Three Dimensional Geometry \u00b7 Straight line: \u00b7 Equation of Line Passing through Two Given Points \u00b7 Cartesian ...","custom_permalink":"study-materials\/three-dimensional-geometry\/class-12-notes\/maths-chapter-11\/"},"categories":[92,93,13,21],"tags":[],"table_tags":[],"acf":[],"yoast_head":"\nThree Dimensional Geometry Class 12 Notes Maths Chapter 11<\/title>\n<meta name=\"description\" content=\"CBSE Class 12 Maths Notes Chapter 11 Three Dimensional Geometry \u00b7 Straight line: \u00b7 Equation of Line Passing through Two 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