Question 1
What are the y- and z-coordinates of a point on the x-axis?
Steps only
P = (x, 0, 0)
Answer
y = 0 and z = 0.
CLASS 11 · MATHEMATICS · CHAPTER 11
Introduction to Three Dimensional Geometry
Locate points in space, calculate distances and use coordinate relationships. Start with the short working and open the explanation for each method.
29 worked answersAll 2 exercise setsAll 4 end-of-chapter questions
29 worked answers · short steps first, explanations when you need them
Exercise 11.1
Textbook pages 211
Question 2
What is the y-coordinate of a point in the XZ-plane?
Steps only
P = (x, 0, z)
Answer
y = 0.
Question 3 · point 1
Which octant contains (1, 2, 3)?
Steps only
Signs of (x, y, z): (+, +, +)
Answer
Octant I.
Question 3 · point 2
Which octant contains (4, -2, 3)?
Steps only
Signs of (x, y, z): (+, −, +)
Answer
Octant IV.
Question 3 · point 3
Which octant contains (4, -2, -5)?
Steps only
Signs of (x, y, z): (+, −, −)
Answer
Octant VIII.
Question 3 · point 4
Which octant contains (4, 2, -5)?
Steps only
Signs of (x, y, z): (+, +, −)
Answer
Octant V.
Question 3 · point 5
Which octant contains (-4, 2, -5)?
Steps only
Signs of (x, y, z): (−, +, −)
Answer
Octant VI.
Question 3 · point 6
Which octant contains (-4, 2, 5)?
Steps only
Signs of (x, y, z): (−, +, +)
Answer
Octant II.
Question 3 · point 7
Which octant contains (-3, -1, 6)?
Steps only
Signs of (x, y, z): (−, −, +)
Answer
Octant III.
Question 3 · point 8
Which octant contains (-2, -4, -7)?
Steps only
Signs of (x, y, z): (−, −, −)
Answer
Octant VII.
Question 4(i)
Name the plane determined by the x- and y-axes.
Steps only
x-axis and y-axis ⇒ XY-plane
Answer
XY-plane.
Question 4(ii)
Give the general coordinates of a point in the XY-plane.
Steps only
P = (x, y, 0)
Answer
(x, y, 0).
Question 4(iii)
Into how many octants do the coordinate planes divide space?
Steps only
2 × 2 × 2 = 8
Answer
8 octants.
Exercise 11.2
Textbook pages 213
Question 1(i)
Find the distance from (2, 3, 5) to (4, 3, 1).
Steps only
d² = (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²
= (2)² + (0)² + (-4)²
= 20
d = 2√5
Answer
2√5 units.
Question 1(ii)
Find the distance from (-3, 7, 2) to (2, 4, -1).
Steps only
d² = (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²
= (5)² + (-3)² + (-3)²
= 43
d = √43
Answer
√43 units.
Question 1(iii)
Find the distance from (-1, 3, -4) to (1, -3, 4).
Steps only
d² = (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²
= (2)² + (-6)² + (8)²
= 104
d = 2√26
Answer
2√26 units.
Question 1(iv)
Find the distance from (2, -1, 3) to (-2, 1, 3).
Steps only
d² = (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²
= (-4)² + (2)² + (0)²
= 20
d = 2√5
Answer
2√5 units.
Question 2
Show that P(−2,3,5), Q(1,2,3) and R(7,0,−1) lie on one line.
Steps only
PQ = √14
QR = 2√14
PR = 3√14
PQ + QR = PR
Answer
The points are collinear, with Q between P and R.
Question 3(i)
Verify that A(0,7,−10), B(1,6,−6), C(4,9,−6) form an isosceles triangle.
Steps only
AB² = 1 + 1 + 16 = 18
BC² = 9 + 9 + 0 = 18
AC² = 16 + 4 + 16 = 36
Answer
AB = BC = 3√2; the triangle is isosceles.
Question 3(ii)
Verify that A(0,7,10), B(−1,6,6), C(−4,9,6) form a right triangle.
Steps only
AB² = 18; BC² = 18; AC² = 36
AB² + BC² = AC²
Answer
The angle at B is 90°.
Question 3(iii)
Verify that A(−1,2,1), B(1,−2,5), C(4,−7,8), D(2,−3,4) form a parallelogram.
Steps only
Midpoint of AC = (3/2, −5/2, 9/2)
Midpoint of BD = (3/2, −5/2, 9/2)
Answer
ABCD is a parallelogram.
Question 4
Find the equation for points equidistant from (1,2,3) and (3,2,−1).
Steps only
(x − 1)² + (y − 2)² + (z − 3)² = (x − 3)² + (y − 2)² + (z + 1)²
4x − 8z = 0
x − 2z = 0
Answer
x − 2z = 0.
Question 5
Find the locus of P whose distances from (4,0,0) and (−4,0,0) add to 10.
Steps only
PA + PB = 10
PB² − PA² = 16x
PB − PA = 8x/5
PA = 5 − 4x/5
(x − 4)² + y² + z² = (5 − 4x/5)²
x²/25 + (y² + z²)/9 = 1
Answer
x²/25 + y²/9 + z²/9 = 1.
Miscellaneous Exercise
Textbook pages 215
Question 1
For consecutive vertices A(3,−1,2), B(1,2,−4), C(−1,1,2), find D.
Steps only
A + C = B + D
D = A + C − B
= (1, −2, 8)
Answer
D = (1, −2, 8).
Question 2 · median from A
Find the median from A in the triangle A(0,0,6), B(0,4,0), C(6,0,0).
Steps only
Opposite-side midpoint = (3, 2, 0)
Median² = 49
Median = 7
Answer
7 units.
Question 2 · median from B
Find the median from B in the triangle A(0,0,6), B(0,4,0), C(6,0,0).
Steps only
Opposite-side midpoint = (3, 0, 3)
Median² = 34
Median = √34
Answer
√34 units.
Question 2 · median from C
Find the median from C in the triangle A(0,0,6), B(0,4,0), C(6,0,0).
Steps only
Opposite-side midpoint = (0, 2, 3)
Median² = 49
Median = 7
Answer
7 units.
Question 3
The centroid of P(2a,2,6), Q(−4,3b,−10), R(8,14,2c) is (0,0,0). Find a, b and c.
Steps only
2a − 4 + 8 = 0 ⇒ a = −2
2 + 3b + 14 = 0 ⇒ b = −16/3
6 − 10 + 2c = 0 ⇒ c = 2
Answer
a = −2; b = −16/3; c = 2.
Question 4
For A(3,4,5) and B(−1,3,−7), find the locus satisfying PA² + PB² = k².
Steps only
(x − 3)² + (y − 4)² + (z − 5)² + (x + 1)² + (y − 3)² + (z + 7)² = k²
2x² + 2y² + 2z² − 4x − 14y + 4z + 109 = k²
Answer
2x² + 2y² + 2z² − 4x − 14y + 4z + 109 = k².