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JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Introduction and Snapshot of Three Dimensional Geometry - Study ref# 929

Introduction In this chapter, we shall study the direction cosines and direction ratios of a line joining two points and also discuss about the equations of lines and planes in space under different conditions, angle between two lines, tw... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Introduction and Snapshot of Three Dimensional Geometry - Study ref# 930

Following sections must be covered to understand the Three Dimensional Geometry: 1. Direction Cosines and Direction Ratios of a Line 2. Equation of a Line in Space 3. Angle between Two Lines 4. Shortest Distance between Two Lines... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Target Setup for Three Dimensional Geometry - Study ref# 931

Student requires to achieve the following milestones to get better grip over Three Dimensional Geometry. The milestones / targets are as follows: Starting Point: Days 0Test TypeAppear on TestTarget TestsScorePractice Test000%Chap... read more

JEE-Main - Grade 11 - Mathematics - Straight Lines - Importance and Snapshot of Straight Lines - Study ref# 629

Introduction: We are familiar with two-dimensional coordinate geometry from earlier classes. Mainly, it is a combination of algebra and geometry. A systematic study of geometry by the use of algebra was first carried out by celebrated ... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Importance and Snapshot of Introduction to Three Dimensional Geometry - Study ref# 635

Introduction You may recall that to locate the position of a point in a plane, we need two intersecting mutually perpendicular lines in the plane. These lines are called the coordinate axes and the two numbers are called the coordinates o... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Importance and Snapshot of Introduction to Three Dimensional Geometry - Study ref# 636

Following section must be covered to understand the Three Dimensional Geometry: 1. Coordinate Axes and Coordinate Planes in Three Dimensional Space 2. Coordinates of a Point in Space 3. Distance between Two Points 4. Section Form... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Target Setup for Introduction to Three Dimensional Geometry - Study ref# 637

Student requires to achieve the following milestones to get better grip over Three Dimensional Geometry. The milestones / targets are as follows:-Starting Point: Days 0Test TypeAppear on TestTarget TestsScorePractice Test000%Chapter T... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5316

A point is in the XZ – plane . What can you say about its y-coordinate? Answer: The coordinates of any point in XZ – plane will be (). Thus -coordinate of the point is zero.... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5318

Find the distance between the following pairs of points: (i) (2,3,5) and (4,3,1) (ii) (-3,7,2) and (2,4,-1) (iii) (-1,3,-4) and (1,-3,4) (iv) (2,-1,3) and (-2,1,3) Answer: (i) The distance PQ between the points P(2,3,5) and Q(4,3,1)... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5319

Show that the points (-2,3,5),(1,2,3) and (7,0,-1) are collinear. Answer: Let A(-2,3,5), B(1,2,3) and C(7,0,-1) be three given points. Then AB =                   = units BC =     ... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5320

Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1). Answer: Let A be any point which is equidistant from points B and C  Then AB =         ... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5321

Find the coordinate of the point which divides the line segment joining the points (-2,3,5) and (1,-4,6) in the ratio (i) 2:3 internally, (ii) 2:3 externally. Answer: (i) Let  be any point which divides the line segment joining the poi... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5322

Given that P(3,2,-4) ,Q(5,4,-6) and R(9,8,-10) are collinear. Find the ratio in which Q divides PR. Answer: Let Q(5,4,-6) divides the line segment joining the points P(3,2,-4) and R(9,8,-10) in the ratio k:1 internally. ∴ Then co... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5323

Find the lengths of the medians of the triangle with vertices A(0,0,6),B(0,4,0) and C(6,0,0). Answer: Here A(0,0,6),B(0,4,0) and C(6,0,0) are the vertices of ∆ABC. Now D is mid-point of BC. ∴ Coordinates of D are    ... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5324

A point R with x-coordinate 4 lies on the line segment joining the points P(2,-3,4) and Q(8,0,10) . Find the coordinates of the point R. [Hint: Suppose R divides PQ in the ratio k:1. The coordinates of the point R are given by read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5325

How far apart are the points (2,0,0) and (-3,0,0)? Answer: Distance between (2,0,0) and (-3,0,0) = units... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5326

Show that if  , then the point  is at a distance 1 unit from the origin. Answer: We have given , ∴Distance of the point from (0,0,0) is       unit Hence, given point is at a distance of 1 unit from the origin.... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5327

Three consecutive vertices of a parallelogram ABCD are (6,-2,4) ,B(2,4,-8) and C(-2,2,4). Find the coordinates of the fourth vertex. [Hint: Diagonals of a parallelogram have the same mid-point] Answer: Let the coordinates of the fourth verte... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5328

The mid-point of the sides of a triangle are (5, 7, 11), (0,8,5) and (2,3,-1). Find its vertices. Answer: Let vertices of ∆ABC be A B   and C and mid-points sides AB, BC and AC are respectively F (2,3,-1), D(5,7,11) and E(0,8,5). ... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5329

The distance of point P(3,4,5) from the yz-plane is (A) 3 units                                  (B) 4 units (C) 5 units                                   (D) 550 units Answer: Given , point P is (3,4,5)  units                         ... read more

JEE-Main - Grade 11 - Mathematics - Introduction to Three Dimensional Geometry - Study ref# 5330

If the mid-points of the sides of a triangle AB, BC and CA are D(1,2,-3), E(3,0,1) and F(-1,1,-4), then the centroid of the ∆ABC is Answer: Consider the vertices of ∆ABC be A,B,C If D is the mid-point of AB, then by using mi... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9608

If a line makes angles 90, 135, 45with the x, y and z –axes respectively, find its direction cosines. Answer: Required direction cosines are <cos90, cos135, cos45> i.e., <0, read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9609

If a line has direction ratios -18, 12, -4, then what are its direction cosines? Answer: Given direction ratios are < -18, 12, -4> As   Therefore, d.c., of the line are  ... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9611

Show that the three lines with direction cosines  are mutually perpendicular. Answer: Let the lines, whose direction cosines are given be and. Let be the angle between , then read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9612

Show that the line through the points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6). Answer: Let the given points be  A(1, -1, 2), B(3, 4, -2), C(0, 3, 2) and D(3, 5, 6). The direction rati... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9613

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector   Answer: We have, Vector equation... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9614

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and is in the direction Answer: We have, read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9615

The Cartesian equation of a line is  write its vector form. Answer: The given Cartesian equation (in symmetrical form) is    the lin... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9616

Find the vector and the Cartesian equations of the lines that passes through the origin and (5, -2, 3). Answer: Let andbe the position vectors of point A= (0, 0, 0) and B= (5, -2, 3) Let read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9617

Find the angle between the following pairs of lines: (i) and   (ii) and read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9618

Find the values of p so that the lines and are at right angles. Answer: Equations of the given lines can be written as and Dir... read more

JEE-Main - Grade 12 - Mathematics - Three Dimensional Geometry - Study ref# 9619

Find the shortest distance between the lines and   Answer: Comparing the given lines by and Answer: The shortest distance between the lines  and is is read more

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