Page 1 Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 1 of 17 Exercise 12.1 Question 1: A point is on the x-axis. What are its y-coordinates and z-coordinates? Answer If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Question 2: A point is in the XZ-plane. What can you say about its y-coordinate? Answer If a point is in the XZ plane, then its y-coordinate is zero. Question 3: Name the octants in which the following points lie: (1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5), (â€“3, â€“1, 6), (2, â€“4, â€“7) Answer The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octant I. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octant IV. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive, positive, and negative respectively. Therefore, this point lies in octant V. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative, positive, and negative respectively. Therefore, this point lies in octant VI. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative, positive, and positive respectively. Therefore, this point lies in octant II. The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative, negative, and positive respectively. Therefore, this point lies in octant III. The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. Page 2 Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 1 of 17 Exercise 12.1 Question 1: A point is on the x-axis. What are its y-coordinates and z-coordinates? Answer If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Question 2: A point is in the XZ-plane. What can you say about its y-coordinate? Answer If a point is in the XZ plane, then its y-coordinate is zero. Question 3: Name the octants in which the following points lie: (1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5), (â€“3, â€“1, 6), (2, â€“4, â€“7) Answer The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octant I. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octant IV. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive, positive, and negative respectively. Therefore, this point lies in octant V. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative, positive, and negative respectively. Therefore, this point lies in octant VI. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative, positive, and positive respectively. Therefore, this point lies in octant II. The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative, negative, and positive respectively. Therefore, this point lies in octant III. The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 2 of 17 Question 4: Fill in the blanks: Answer (i) The x-axis and y-axis taken together determine a plane known as . (ii) The coordinates of points in the XY-plane are of the form . (iii) Coordinate planes divide the space into octants. Page 3 Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 1 of 17 Exercise 12.1 Question 1: A point is on the x-axis. What are its y-coordinates and z-coordinates? Answer If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Question 2: A point is in the XZ-plane. What can you say about its y-coordinate? Answer If a point is in the XZ plane, then its y-coordinate is zero. Question 3: Name the octants in which the following points lie: (1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5), (â€“3, â€“1, 6), (2, â€“4, â€“7) Answer The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octant I. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octant IV. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive, positive, and negative respectively. Therefore, this point lies in octant V. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative, positive, and negative respectively. Therefore, this point lies in octant VI. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative, positive, and positive respectively. Therefore, this point lies in octant II. The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative, negative, and positive respectively. Therefore, this point lies in octant III. The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 2 of 17 Question 4: Fill in the blanks: Answer (i) The x-axis and y-axis taken together determine a plane known as . (ii) The coordinates of points in the XY-plane are of the form . (iii) Coordinate planes divide the space into octants. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 3 of 17 Exercise 12.2 Question 1: Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1) (ii) (â€“3, 7, 2) and (2, 4, â€“1) (iii) (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) (2, â€“1, 3) and (â€“2, 1, 3) Answer The distance between points P(x 1 , y 1 , z 1 ) and P(x 2 , y 2 , z 2 ) is given by (i) Distance between points (2, 3, 5) and (4, 3, 1) (ii) Distance between points (â€“3, 7, 2) and (2, 4, â€“1) (iii) Distance between points (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) Distance between points (2, â€“1, 3) and (â€“2, 1, 3) Page 4 Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 1 of 17 Exercise 12.1 Question 1: A point is on the x-axis. What are its y-coordinates and z-coordinates? Answer If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Question 2: A point is in the XZ-plane. What can you say about its y-coordinate? Answer If a point is in the XZ plane, then its y-coordinate is zero. Question 3: Name the octants in which the following points lie: (1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5), (â€“3, â€“1, 6), (2, â€“4, â€“7) Answer The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octant I. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octant IV. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive, positive, and negative respectively. Therefore, this point lies in octant V. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative, positive, and negative respectively. Therefore, this point lies in octant VI. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative, positive, and positive respectively. Therefore, this point lies in octant II. The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative, negative, and positive respectively. Therefore, this point lies in octant III. The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 2 of 17 Question 4: Fill in the blanks: Answer (i) The x-axis and y-axis taken together determine a plane known as . (ii) The coordinates of points in the XY-plane are of the form . (iii) Coordinate planes divide the space into octants. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 3 of 17 Exercise 12.2 Question 1: Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1) (ii) (â€“3, 7, 2) and (2, 4, â€“1) (iii) (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) (2, â€“1, 3) and (â€“2, 1, 3) Answer The distance between points P(x 1 , y 1 , z 1 ) and P(x 2 , y 2 , z 2 ) is given by (i) Distance between points (2, 3, 5) and (4, 3, 1) (ii) Distance between points (â€“3, 7, 2) and (2, 4, â€“1) (iii) Distance between points (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) Distance between points (2, â€“1, 3) and (â€“2, 1, 3) Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 4 of 17 Question 2: Show that the points (â€“2, 3, 5), (1, 2, 3) and (7, 0, â€“1) are collinear. Answer Let points (â€“2, 3, 5), (1, 2, 3), and (7, 0, â€“1) be denoted by P, Q, and R respectively. Points P, Q, and R are collinear if they lie on a line. Page 5 Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 1 of 17 Exercise 12.1 Question 1: A point is on the x-axis. What are its y-coordinates and z-coordinates? Answer If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Question 2: A point is in the XZ-plane. What can you say about its y-coordinate? Answer If a point is in the XZ plane, then its y-coordinate is zero. Question 3: Name the octants in which the following points lie: (1, 2, 3), (4, â€“2, 3), (4, â€“2, â€“5), (4, 2, â€“5), (â€“4, 2, â€“5), (â€“4, 2, 5), (â€“3, â€“1, 6), (2, â€“4, â€“7) Answer The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive. Therefore, this point lies in octant I. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, 3) are positive, negative, and positive respectively. Therefore, this point lies in octant IV. The x-coordinate, y-coordinate, and z-coordinate of point (4, â€“2, â€“5) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, â€“5) are positive, positive, and negative respectively. Therefore, this point lies in octant V. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, â€“5) are negative, positive, and negative respectively. Therefore, this point lies in octant VI. The x-coordinate, y-coordinate, and z-coordinate of point (â€“4, 2, 5) are negative, positive, and positive respectively. Therefore, this point lies in octant II. The x-coordinate, y-coordinate, and z-coordinate of point (â€“3, â€“1, 6) are negative, negative, and positive respectively. Therefore, this point lies in octant III. The x-coordinate, y-coordinate, and z-coordinate of point (2, â€“4, â€“7) are positive, negative, and negative respectively. Therefore, this point lies in octant VIII. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 2 of 17 Question 4: Fill in the blanks: Answer (i) The x-axis and y-axis taken together determine a plane known as . (ii) The coordinates of points in the XY-plane are of the form . (iii) Coordinate planes divide the space into octants. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 3 of 17 Exercise 12.2 Question 1: Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1) (ii) (â€“3, 7, 2) and (2, 4, â€“1) (iii) (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) (2, â€“1, 3) and (â€“2, 1, 3) Answer The distance between points P(x 1 , y 1 , z 1 ) and P(x 2 , y 2 , z 2 ) is given by (i) Distance between points (2, 3, 5) and (4, 3, 1) (ii) Distance between points (â€“3, 7, 2) and (2, 4, â€“1) (iii) Distance between points (â€“1, 3, â€“4) and (1, â€“3, 4) (iv) Distance between points (2, â€“1, 3) and (â€“2, 1, 3) Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 4 of 17 Question 2: Show that the points (â€“2, 3, 5), (1, 2, 3) and (7, 0, â€“1) are collinear. Answer Let points (â€“2, 3, 5), (1, 2, 3), and (7, 0, â€“1) be denoted by P, Q, and R respectively. Points P, Q, and R are collinear if they lie on a line. Class XI Chapter 12 â€“ Introduction to Three Dimensional Geometry Maths Page 5 of 17 Here, PQ + QR = PR Hence, points P(â€“2, 3, 5), Q(1, 2, 3), and R(7, 0, â€“1) are collinear. Question 3: Verify the following: (i) (0, 7, â€“10), (1, 6, â€“6) and (4, 9, â€“6) are the vertices of an isosceles triangle. (ii) (0, 7, 10), (â€“1, 6, 6) and (â€“4, 9, 6) are the vertices of a right angled triangle. (iii) (â€“1, 2, 1), (1, â€“2, 5), (4, â€“7, 8) and (2, â€“3, 4) are the vertices of a parallelogram. Answer (i) Let points (0, 7, â€“10), (1, 6, â€“6), and (4, 9, â€“6) be denoted by A, B, and C respectively. Here, AB = BC ? CA Thus, the given points are the vertices of an isosceles triangle. (ii) Let (0, 7, 10), (â€“1, 6, 6), and (â€“4, 9, 6) be denoted by A, B, and C respectively.Read More

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