Mathematics · Textbook solutions

Introduction to Three Dimensional Geometry

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 29 questions

11.2-11.3 Coordinate Axes, Planes and Points in Space

8 q

Solved Examples

Worked · 2
  1. 11.1 Eg.1
    In Fig 11.3, if P is (2,4,5)(2, 4, 5), find the coordinates of F.
  2. 11.1 Eg.2
    Find the octant in which the points (−3,1,2)(-3, 1, 2) and (−3,1,−2)(-3, 1, -2) lie.

Exercise 11.1

Practice · 6
  1. Ex 11.1 Q1
    A point is on the xx-axis. What are its yy-coordinate and zz-coordinates?
  2. Ex 11.1 Q2
    A point is in the XZ-plane. What can you say about its yy-coordinate?
  3. Ex 11.1 Q3
    Name the octants in which the following points lie: (1,2,3)(1, 2, 3), (4,−2,3)(4, -2, 3), (4,−2,−5)(4, -2, -5), (4,2,−5)(4, 2, -5), (−4,2,−5)(-4, 2, -5), (−4,2,5)(-4, 2, 5), (−3,−1,6)(-3, -1, 6) (−2,−4,−7)(-2, -4, -7).
  4. Fill in the blanks:
    Ex 11.1 Q4(i)
    The xx-axis and yy-axis taken together determine a plane known as________.
  5. Ex 11.1 Q4(ii)
    The coordinates of points in the XY-plane are of the form _______.
  6. Ex 11.1 Q4(iii)
    Coordinate planes divide the space into ______ octants.

11.4 Distance between Two Points

14 q

Solved Examples

Worked · 4
  1. 11.2 Eg.3
    Find the distance between the points P(1,−3,4)P(1, -3, 4) and Q(−4,1,2)Q(-4, 1, 2).
  2. 11.2 Eg.4
    Show that the points P(−2,3,5)P(-2, 3, 5), Q(1,2,3)Q(1, 2, 3) and R(7,0,−1)R(7, 0, -1) are collinear.
  3. 11.2 Eg.5
    Are the points A(3,6,9)A(3, 6, 9), B(10,20,30)B(10, 20, 30) and C(25,−41,5)C(25, -41, 5), the vertices of a right angled triangle?
  4. 11.2 Eg.6
    Find the equation of set of points P such that PA2+PB2=2k2PA^2 + PB^2 = 2k^2, where A and B are the points (3,4,5)(3, 4, 5) and (−1,3,−7)(-1, 3, -7), respectively.

Exercise 11.2

Practice · 10
  1. Find the distance between the following pairs of points:
    Ex 11.2 Q1(i)
    (2,3,5)(2, 3, 5) and (4,3,1)(4, 3, 1)
  2. Ex 11.2 Q1(ii)
    (−3,7,2)(-3, 7, 2) and (2,4,−1)(2, 4, -1)
  3. Ex 11.2 Q1(iii)
    (−1,3,−4)(-1, 3, -4) and (1,−3,4)(1, -3, 4)
  4. Ex 11.2 Q1(iv)
    (2,−1,3)(2, -1, 3) and (−2,1,3)(-2, 1, 3).
  5. Ex 11.2 Q2
    Show that the points (−2,3,5)(-2, 3, 5), (1,2,3)(1, 2, 3) and (7,0,−1)(7, 0, -1) are collinear.
  6. Verify the following:
    Ex 11.2 Q3(i)
    (0,7,−10)(0, 7, -10), (1,6,−6)(1, 6, -6) and (4,9,−6)(4, 9, -6) are the vertices of an isosceles triangle.
  7. Ex 11.2 Q3(ii)
    (0,7,10)(0, 7, 10), (−1,6,6)(-1, 6, 6) and (−4,9,6)(-4, 9, 6) are the vertices of a right angled triangle.
  8. Ex 11.2 Q3(iii)
    (−1,2,1)(-1, 2, 1), (1,−2,5)(1, -2, 5), (4,−7,8)(4, -7, 8) and (2,−3,4)(2, -3, 4) are the vertices of a parallelogram.
  9. Ex 11.2 Q4
    Find the equation of the set of points which are equidistant from the points (1,2,3)(1, 2, 3) and (3,2,−1)(3, 2, -1).
  10. Ex 11.2 Q5
    Find the equation of the set of points P, the sum of whose distances from A(4,0,0)A(4, 0, 0) and B(−4,0,0)B(-4, 0, 0) is equal to 10.

Miscellaneous Exercise on Chapter 11

7 q

Miscellaneous Examples

Worked · 3
  1. Misc Eg.7
    Show that the points A(1,2,3)A(1, 2, 3), B(−1,−2,−1)B(-1, -2, -1), C(2,3,2)C(2, 3, 2) and D(4,7,6)D(4, 7, 6) are the vertices of a parallelogram ABCD, but it is not a rectangle.
  2. Misc Eg.8
    Find the equation of the set of the points P such that its distances from the points A(3,4,−5)A(3, 4, -5) and B(−2,1,4)B(-2, 1, 4) are equal.
  3. Misc Eg.9
    The centroid of a triangle ABC is at the point (1,1,1)(1, 1, 1). If the coordinates of A and B are (3,−5,7)(3, -5, 7) and (−1,7,−6)(-1, 7, -6), respectively, find the coordinates of the point C.

Miscellaneous Exercise

Practice · 4
  1. Misc Q1
    Three vertices of a parallelogram ABCD are A(3,−1,2)A(3, -1, 2), B(1,2,−4)B(1, 2, -4) and C(−1,1,2)C(-1, 1, 2). Find the coordinates of the fourth vertex.
  2. Misc Q2
    Find the lengths of the medians of the triangle with vertices A(0,0,6)A(0, 0, 6), B(0,4,0)B(0, 4, 0) and (6,0,0)(6, 0, 0).
  3. Misc Q3
    If the origin is the centroid of the triangle PQR with vertices P(2a,2,6)P(2a, 2, 6), Q(−4,3b,−10)Q(-4, 3b, -10) and R(8,14,2c)R(8, 14, 2c), then find the values of aa, bb and cc.
  4. Misc Q4
    If A and B be the points (3,4,5)(3, 4, 5) and (−1,3,−7)(-1, 3, -7), respectively, find the equation of the set of points P such that PA2+PB2=k2PA^2 + PB^2 = k^2, where kk is a constant.