 # Ch11. Three Dimensional Geometry

## Achievements

The following badges can be earned while learning.   ## Certificate

Certificate on successful completion of this course. In this online course, you will learn direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane. For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that have questions from NCERT Textbook exercise, NCERT Examples, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course. Please note, this course comprises of live session recordings of Ashish Kumar (Agam Sir). Please ignore, if you have heard Agam Sir discussing about live session or doubts session in the course. Live sessions or doubts sessions are no longer offered.

## Course Content

The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Lecture – 1

Question 1. If a line makes angle {\displaystyle 90^\circ ,60^\circ } and {\displaystyle 30^\circ } with the positive direction of x, y and z-axis respectively, find its direction cosines.
Question 2. If a line has direction ratios 2, −1, −2 determine its direction cosines.
Question 3. Find the direction cosines of the line passes through the two points (−2, 4,−5) and (1, 2, 3).
Question 4. Find the direction cosines of x, y and z-axis.
Question 5. Show that the points A(2, 3, −4), B(1, −2, 3) and C(3, 8, −11) are collinear.
Question 6. Find the direction cosines of the sides of the triangle whose vertices are (3, 5,−4), (−1, 1, 2) and (−5, −5, −2).
Question 7. Show that the three lines with the direction cosines {\displaystyle \frac{{12}}{{13}},\frac{{ - 3}}{{13}},\frac{{ - 4}}{{13}};\,\,\frac{4}{{13}},\frac{{12}}{{13}},\frac{3}{{13}};\,\,\frac{3}{{13}},\frac{{ - 4}}{{13}},\frac{{12}}{{13}}} are mutually perpendicular.
Question 8. Show that the line through the points {\displaystyle (1, - 1, 2), (3,4, - 2)} is perpendicular to the line through the points {\displaystyle (0, 3, 2) , (3, 5, 6) } .
Question 9. Show that the line through the points {\displaystyle (4,\,\,7,\,\,8),\,\,(2,\,\,3,\,\,4)} is parallel to the line through the points {\displaystyle ( - 1, - 2,\,\,\,1),\,\,(1,\,\,2,\,\,5)}
Question 10. Find the angle between the lines whose direction ratios are a, b, c and {\displaystyle b - c,\,\,c - a,\,\,a - b} .
Question 11. If the coordinates of the points A, B, C, D be {\displaystyle (1,\,\,2,\,\,3),\,\,(4,\,\,5,\,\,7),\,\,( - 4,\,\,3,\,\, - 6)} and {\displaystyle (2,\,\,9,\,\,2)} respectively, then find the angle between the lines AB and CD.

### Lecture – 2

Question 1. Find the Cartesian and Vector equations of the line through the point (5, 2, -4) and which is parallel to the vector {\displaystyle 3\hat i + 2\hat j - 8\hat k} .
Question 2. Find the vector equation for the line passing through the points (−1,0,2) and (3, 4, 6).
Question 3. Find the equation of the line in vector and in cartesian form that passes through the point with position vector {\displaystyle 2\hat i - \hat j + 4\hat k} and is in the direction {\displaystyle \hat i + 2\hat j - \hat k} .
Question 4. The Cartesian equation of the line is {\displaystyle \frac{{x - 5}}{3} = \frac{{y - 4}}{7} = \frac{{z - 6}}{2}} . Write its vector form.
Question 5. Find the vector and the Cartesian equations of the lines that passes through the origin and (5, −2, 3).
Question 6. Find the Cartesian and the vector equation of the line that passes through the points (3,−2,−5), (3,−2,6).
Question 7. Find the angle between following pairs of lines:
a. {\displaystyle \vec r = 3\hat i + 2\hat j - 4\hat k + \lambda (\hat i + 2\hat j + 2\hat k)\,\,\,\,\& \,\,\,\,\,\vec r = 5\hat i - 2\hat j + \mu (3\hat i + 2\hat j + 6\hat k)}
b. {\displaystyle \vec r = \hat i - 5\hat j + \hat k + \lambda (3\hat i + 2\hat j + 6\hat k)\,\,\,\,\& \,\,\,\,\,\vec r = 7\hat i - 6\hat k + \mu (\hat i + 2\hat j + 2\hat k)}
c. {\displaystyle \vec r = 3\hat i + \hat j - 2\hat k + \lambda (\hat i - \hat j - 2\hat k)\,\,\,\,\& \,\,\,\,\,\vec r = 2\hat i - \hat j - 56\hat k + \mu (3\hat i - 5\hat j - 4\hat k)}
d. {\displaystyle \frac{{x - 2}}{2} = \frac{{y - 1}}{5} = \frac{{z + 3}}{{ - 3}}\,\,\,\,\,\& \,\,\,\,\frac{{x + 2}}{{ - 1}} = \frac{{y - 4}}{8} = \frac{{z - 5}}{4}}
e. {\displaystyle \frac{x}{2} = \frac{y}{2} = \frac{z}{1}\,\,\,\,\& \,\,\,\frac{{x - 5}}{4} = \frac{{y - 2}}{1} = \frac{{z - 3}}{8}}
f. {\displaystyle \frac{{x + 3}}{3} = \frac{{y - 1}}{5} = \frac{{z + 3}}{4}\,\,\,\,\,\,\& \,\,\,\,\frac{{x + 1}}{1} = \frac{{y - 4}}{1} = \frac{{z - 5}}{2}}
Question 8. Find the values of p so that the lines {\displaystyle \frac{{1 - x}}{3} = \frac{{7y - 14}}{{2p}} = \frac{{z - 3}}{2}} and {\displaystyle \frac{{7 - 7x}}{{3p}} = \frac{{y - 5}}{1} = \frac{{6 - z}}{5}} are at right angles.
Question 9. Show that the lines {\displaystyle \frac{{x - 5}}{7} = \frac{{y + 2}}{{ - 5}} = \frac{z}{1}} and {\displaystyle \frac{x}{1} = \frac{y}{2} = \frac{z}{3}} are perpendicular to each other.
Question 10. If the lines {\displaystyle \frac{{x - 1}}{{ - 3}} = \frac{{y - 2}}{{2k}} = \frac{{z - 3}}{2}} and {\displaystyle \frac{{x - 1}}{{3k}} = \frac{{y - 1}}{1} = \frac{{z - 6}}{{ - 5}}} are perpendicular, find the value of k.
Question 11. Find the equation of the line passing through P (−1, 3, −2) and perpendicular to the lines {\displaystyle \frac{x}{1} = \frac{y}{2} = \frac{z}{3}} and {\displaystyle \frac{{x + 2}}{{ - 3}} = \frac{{y - 1}}{2} = \frac{{z + 1}}{5}} .
Question 12. Find the vector and cartesian equation of the lines passing through the points P(0, 1, −2) and Q (−1, −1, −3). Also prove that it passes through the point R whose position vector is {\displaystyle - 3\hat i - 5\hat j - 5\hat k} .
Question 13. Find the equation of the line drawn perpendicular from the point P(1, 6, 3) to the line {\displaystyle \frac{x}{1} = \frac{{y - 1}}{2} = \frac{{z - 2}}{3}} . Also find the perpendicular distance of the given line from the point.
Question 14. Find the point on the line {\displaystyle \frac{{x + 2}}{3} = \frac{{y + 1}}{2} = \frac{{z - 3}}{2}} at a distance {\displaystyle 3\sqrt 2 } from the point (1, 2, 3).
Question 15. Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line {\displaystyle \frac{{x - 11}}{{10}} = \frac{{y + 2}}{{ - 4}} = \frac{{z + 8}}{{ - 11}}} .
Question 16. Find the points on the line {\displaystyle \frac{{x + 2}}{3} = \frac{{y + 1}}{2} = \frac{{z - 3}}{2}} at a distance 5 units from the point P(1, 3, 3).
Question 17. Find the angle between following pairs of lines {\displaystyle \frac{{ - x + 2}}{{ - 2}} = \frac{{y - 1}}{7} = \frac{{z + 3}}{{ - 3}}\,\,\,\,\& \,\,\,\frac{{x + 2}}{{ - 1}} = \frac{{2y - 8}}{4} = \frac{{z - 5}}{4}} and check whether the lines are parallel or perpendicular.
Question 18. Find the vector and the cartesian equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines {\displaystyle \frac{{x - 8}}{3} = \frac{{y + 19}}{{ - 16}} = \frac{{z - 10}}{7}} and {\displaystyle \frac{{x - 15}}{3} = \frac{{y - 29}}{8} = \frac{{z - 5}}{{ - 5}}} .
Question 19. Find the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines {\displaystyle \bar r = (\hat i + \hat j - \hat k) + \lambda (2\hat i - 2\hat j + \hat k)} and {\displaystyle \vec r = (2\hat i - \hat j - 3\hat k) + \mu (\hat i + 2\hat j + 2\hat k)} .
Question 20. Find the perpendicular distance of point (2, 3, 4) from the line {\displaystyle \frac{{4 - x}}{2} = \frac{y}{6} = \frac{{1 - z}}{3}} .
Question 21. Find the image of the point (1, 6, 3) in the line {\displaystyle \frac{x}{1} = \frac{{y - 1}}{2} = \frac{{z - 2}}{3}} .
Question 22. Let the point P (5, 9, 3) lies on the top of the Qutub Minar, Delhi. Find the image of the point on the line {\displaystyle \frac{{x - 1}}{2} = \frac{{y - 2}}{3} = \frac{{z - 3}}{4}} . Do you think that the conservation of monuments is important and why?
Question 23. Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).

### Lecture – 3

Question 1. Find the shortest distance between the lines & check whether lines are intersecting or not:
a. {\displaystyle \vec r = (\hat i + 2\hat j + \hat k) + \lambda (\hat i - \hat j + \hat k){\mkern 1mu} {\mkern 1mu} \& {\mkern 1mu} {\mkern 1mu} \vec r = (2\hat i - \hat j - \hat k) + \mu (2\hat i + \hat j + 2\hat k)}
b. {\displaystyle \frac{{x + 1}}{7} = \frac{{y + 1}}{{ - 6}} = \frac{{z + 1}}{1}\,\,\,\,\& \,\,\,\frac{{x - 3}}{1} = \frac{{y - 5}}{{ - 2}} = \frac{{z - 7}}{1}}
c. {\displaystyle \vec r = (\hat i + 2\hat j + 3\hat k) + \lambda (\hat i - 3\hat j + 2\hat k)\,\,\,\,\& \,\,\,\,\vec r = 4\hat i + 5\hat j + 6\hat k + \mu (2\hat i + 3\hat j + \hat k)}
d. {\displaystyle \vec r = (1 - t)\hat i + (t - 2)\hat j + (3 - 2t)\hat k\,\,\& \,\,\vec r = (s + 1)\hat i + (2s - 1)\hat j - (2s + 1)\hat k}
e. {\displaystyle \vec r = (6\hat i + 2\hat j + 2\hat k) + \lambda (\hat i - 2\hat j + 2\hat k)\,\,\,\& \,\,\,\vec r = - 4\hat i - \hat k + \mu (3\hat i - 2\hat j - 2\hat k)}
Question 2. Find the shortest distance between the lines and hence write whether the lines are intersecting or not : {\displaystyle \frac{{x - 1}}{2} = \frac{{y + 1}}{3} = z\,\,\,\,\,\,\& \,\,\,\,\,\frac{{x + 1}}{5} = \frac{{y - 2}}{1} = \frac{z}{2}} .
Question 3. Show that the lines {\displaystyle \frac{{x + 3}}{{ - 3}} = \frac{{y - 1}}{1} = \frac{{z - 5}}{5}\,\,\,\& \,\,\frac{{x + 1}}{{ - 1}} = \frac{{y - 2}}{2} = \frac{{z - 5}}{5}} are coplanar.
Question 4. Show that the lines {\displaystyle \frac{{x - a + d}}{{\alpha - \delta }} = \frac{{y - a}}{\alpha } = \frac{{z - a - d}}{{\alpha + \delta }}\,\,\,\& \,\,\frac{{x - b + c}}{{\beta - \gamma }} = \frac{{y - b}}{\beta } = \frac{{z - b - c}}{{\beta + \gamma }}} are coplanar.
Question 5. Find the distance between lines
a. {\displaystyle \vec r = \hat i + 2\hat j + \hat k + \lambda (2\hat i + 3\hat j + 6\hat k)\,\,\,\,\& \,\,\,\,\vec r = 3\hat i + 3\hat j - 5\hat k + \mu (2\hat i + 3\hat j + 6\hat k)} .
Question 6. Find the distance between following pairs of lines:
a. {\displaystyle \vec r = (\hat i + 2\hat j + 3\hat k) + \lambda (\hat i - \hat j + \hat k)\,\,\,\,\,\& \,\,\,\,\,\,\vec r = (2\hat i - \hat j - \hat k) + \mu ( - \hat i + \hat j - \hat k)}
b. {\displaystyle \vec r = (\hat i + \hat j) + \lambda (2\hat i - \hat j + \hat k)\,\,\,\,\,\& \,\,\,\,\vec r = (2\hat i + \hat j - \hat k) + \mu (4\hat i - 2\hat j + 2\hat k)}

### Lecture – 4

Question 1. Find the vector and the cartesian equations of the plane which passes through the point {\displaystyle (5,2, - 4)} and perpendicular to the line with direction ratios 2, 3, −1.
Question 2. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector {\displaystyle 3\hat i + 5\hat j - 6\hat k} .
Question 3. Find the vector and the cartesian equations of the planes that passes through the point (1, 0, −2) and the normal to the plane is {\displaystyle \hat i + \hat j - \hat k} .
Question 4. Find the vector and the Cartesian equations of the plane that passes through points (1, 1, 0), (1, 2, 1), (−2, 2, −1).
Question 5. Find the equation of the plane passes through the points (1,1,−1), (6,4,−5), (−4,−2, 3).
Question 6. The foot of perpendicular drawn from origin to the plane is (4, −2, −5). Find the equation of the plane.
Question 7. Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes {\displaystyle 2x + 3y - 2z = 5} and {\displaystyle x + 2y - 3z = 8} .
Question 8. Find the equation of the plane passing through {\displaystyle (a,b,c)} and parallel to the plane {\displaystyle \vec r.(\hat i + \hat j + \hat k) = 2} .
Question 9. Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes {\displaystyle x + 2y + 3z = 5} and {\displaystyle 3x + 3y + z = 0} .
Question 10. Find the equation of the plane passing through the points (1, −1, 2) and (2, −2, 2) and which is perpendicular to the plane {\displaystyle 6x - 2y + 2z = 9} .
Question 11. Find the equation of the plane through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane {\displaystyle x + 2y + 2z = 5} .
Question 12. Find the vector equation of the plane which is at a distance of {\displaystyle \frac{6}{{\sqrt {29} }}} from the origin and its normal vector from the origin is {\displaystyle 2\hat i - 3\hat j + 4\hat k} . Also find its cartesian form.
Question 13. Find the direction cosines of the unit vector perpendicular to the plane {\displaystyle \vec r.(6\hat i - 3\hat j - 2\hat k) + 1 = 0} passing through the origin.
Question 14. Find the distance of the plane {\displaystyle 2x - 3y + 4z - 6 = 0} from the origin.
Question 15. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
a. {\displaystyle 7x + 5y + 6z + 30 = 0,\,\,\,\,3x - y - 10z + 4 = 0}
b. {\displaystyle 2x + y + 3z - 2 = 0,\,\,\,\,x - 2y + 5 = 0}
c. {\displaystyle 2x - 2y + 4z + 5 = 0,\,\,\,3x - 3y + 6z - 1 = 0}
d. {\displaystyle \vec r.(2\hat i + 2\hat j - 3\hat k) = 5,\,\,\,\,\,\,\vec r.(3\hat i - 3\hat j + 5\hat k) = 3}
e. {\displaystyle \vec r.(2\hat i - 3\hat j + 4\hat k\,) = 1,\,\,\,\,\vec r.( - \hat i + \hat j\,) = 4}
f. {\displaystyle \vec r.(2\hat i - \hat j + 2\hat k) = 6,\,\,\,\,\,\vec r.(3\hat i + 6\hat j - 2\hat k) = 9}
Question 16. Determine the value of {\displaystyle \lambda } for which the following planes are perpendicular to each other
a. {\displaystyle \vec r.(\hat i + 2\hat j + 3\hat k) = 7,\,\,\,\,\vec r.(\lambda \hat i + 2\hat j - 7\hat k) = 26}
b. {\displaystyle 2x - 4y + 3z = 5,\,\,\,\,x + 2y + \lambda z = 5}
c. {\displaystyle 3x - 6y - 2z = 7,\,\,\,2x + y - \lambda z = 5}
Question 17. Write the equation of the plane whose intercepts on the coordinate axes are 2, -3 and 4.
Question 18. Reduce the equations of the following planes in intercept form and find its intercepts on the coordinate axes:
a. {\displaystyle 4x + 3y - 6z - 12 = 0}
b. {\displaystyle 2x + 3y - z = 6}
c. {\displaystyle 2x - y + z = 5}
Question 19. Find the equation of a plane which meets the axes in A, B and C, given that the centroid of the triangle ABC is the point {\displaystyle (\alpha ,\beta ,\gamma )} .
Question 20. Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.
Question 21. A plane meets the coordinate axes at A, B and C respectively such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.

### Lecture – 5

Question 1. Find the distance of the point {\displaystyle (2,\,5,\, - 3)} from the plane {\displaystyle \vec r.(6\hat i - 3\hat j + 2\hat k) = 4} .
Question 2. In the following cases find the distance of each of the given points from the corresponding given plane:
Point Plane
i. {\displaystyle (3, - 2,1)} {\displaystyle 3x - 4y + 12z = 3}
ii. {\displaystyle (2,3, - 5)} {\displaystyle 2x - y + 2z + 3 = 0}
iii. {\displaystyle ( - 6,0,0)} {\displaystyle 2x - 3y + 6z - 2 = 0}
Question 3. Find the distance between the point P (6, 5, 9) and the plane determined by the points {\displaystyle A(3, - 1,2)} , {\displaystyle B(5,2,4)} and {\displaystyle C( - 1, - 1,6)} .
Question 4. Find the distance between the two planes: {\displaystyle 2x + 3y + 4z = 4} and {\displaystyle 4x + 6y + 8z = 12} .
Question 5. Prove that if a plane has the intercepts a, b , c and is at the distance p units from the origin, then {\displaystyle \frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}} = \frac{1}{{{p^2}}}} .
Question 6. Find the distance of the point whose position vector is {\displaystyle (2\hat i + \hat j - \hat k)} from the plane {\displaystyle \vec r.(\hat i - 2\hat j + 4\hat k) = 9} .
Question 7. Find the distance between the planes {\displaystyle \vec r.(\hat i + 2\hat j + 3\hat k) + 7 = 0} & {\displaystyle \vec r.(2\hat i + 4\hat j + 6\hat k) + 7 = 0}
Question 8. If the points {\displaystyle (1,1,p)} and {\displaystyle ( - 3,0,1)} be equidistant from the plane {\displaystyle \vec r.(3\hat i + 4\hat j - 12\hat k) + 13 = 0} , then find the value of p.
Question 9. Find the angle between the line {\displaystyle \frac{{x + 1}}{2} = \frac{y}{3} = \frac{{z - 3}}{6}} and the plane {\displaystyle 10x + 2y - 11z = 3}
Question 10. Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane {\displaystyle \vec r.(\hat i + 2\hat j - 5\hat k) + 9 = 0.}
Question 11. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.
Question 12. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.
Question 13. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the XY-plane.
Question 14. Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane {\displaystyle 2x + y + z = 7} .
Question 15. If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.
Question 16. Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes {\displaystyle \vec r.(\hat i - \hat j + 2\hat k) = 5} and {\displaystyle \vec r.(3\hat i + \hat j + \hat k) = 6} .
Question 17. Find the distance of the point {\displaystyle ( - 1,\, - 5,\, - 10)} from the point of intersection of the line {\displaystyle \vec r = 2\hat i - \hat j + 2\hat k + \lambda (3\hat i + 4\hat j + 2\hat k)} and the plane {\displaystyle \vec r.(\hat i - \hat j + \hat k) = 5} .
Question 18. Find the coordinates of the point where the line through {\displaystyle (3,\, - 4,\, - 5)} and {\displaystyle (2,\, - 3,\,1)} crosses the plane passing through three points {\displaystyle (2,\,2,\,1)} , {\displaystyle (3,\,0,\,1)} and {\displaystyle (4,\, - 1,\,0)} .
Question 19. If a line makes angles {\displaystyle \alpha ,\,\,\beta ,\,\,\gamma } with the positive direction of the coordinate axes; then find the value of {\displaystyle si{n^2}\alpha + si{n^2}\beta + si{n^2}\gamma } .
Question 20. Prove that the line through A(0, −1, −1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(−4,4, 4).
Question 21. Find the equation of plane which perpendicularly bisect the line joining the points A(2, 3, 4) and B(4, 5, 8).
Question 22. If a line drawn from the point {\displaystyle ( - 2,\, - 1,\, - 3)} meets a plane at right angle at the point {\displaystyle (1,\, - 3,\,3)} , find the equation of the plane.
Question 23. Find the equations of the line passing through the point (3, 0, 1) and parallel to the planes {\displaystyle x + 2y = 0} and {\displaystyle 3y - z = 0} .
Question 24. Show that the lines {\displaystyle \frac{{x + 1}}{3} = \frac{{y + 3}}{5} = \frac{{z + 5}}{7}} and {\displaystyle \frac{{x - 2}}{1} = \frac{{y - 4}}{3} = \frac{{z - 6}}{5}} intersect each other. Find the point of intersection.
Question 25. A vector of magnitude 8 units is inclined to x-axis at {\displaystyle 45^\circ } , y-axis at {\displaystyle 60^\circ } and an acute angle with the z-axis. If a plane passes through a point {\displaystyle (\sqrt 2 ,\,\, - 1,\,\,1)} and is normal to {\displaystyle \vec n} , find its equation in vector form.
Question 26. Show that the line whose vector equation is {\displaystyle \vec r = (2\hat i - 2\hat j + 3\hat k) + \lambda (\hat i - \hat j + 4\hat k)} parallel to the plane whose vector equation is {\displaystyle \vec r.(\hat i + 5\hat j + \hat k) = 5} . Also, find the distance between them.

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