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Math Exercises: Analytic Geometry of the Straight Line and Plane

 

 

 

 

1. Find the direction vector and the normal vector of a straight line p, if :

 

Analytic geometry of the straight line and plane - Exercise 1

 

2. Find the parametric, the general and the slope-intercept equation of a straight line p passing through the points :

 

Analytic geometry of the straight line and plane - Exercise 2

 

3. A straight line p with a direction vector d and a normal vector n passes through the point K. Find the parametric, the general and the slope-intercept equation of a straight line p, if :

 

Analytic geometry of the straight line and plane - Exercise 3

 

4. Find the parametric, the general and the slope-intercept equation of a straight line p, which passes through the point M, if the slope angle between a straight line p and the x-axis is φ :

 

Analytic geometry of the straight line and plane - Exercise 4

 

5. Transform the parametric equations of the straight line into the general equation and into the slope-intercept equation :

 

Analytic geometry of the straight line and plane - Exercise 5

 

6. Transform the general equation of the straight line into the parametric equations and into the slope-intercept equation :

 

Analytic geometry of the straight line and plane - Exercise 6

 

7. Transform the slope-intercept equation of the straight line into the parametric equations and into the general equation :

 

Analytic geometry of the straight line and plane - Exercise 7

 

8. Find the general equation and the slope-intercept equation of a straight line, which passes through the point L and which is parallel to the given straight line p :

 

Analytic geometry of the straight line and plane - Exercise 8

 

9. Find the general equation and the slope-intercept equation of a straight line, which passes through the point N and which is perpendicular to the given straight line p :

 

Analytic geometry of the straight line and plane - Exercise 9

 

10. Find the slope angle of a straight line given by the equation :

 

Analytic geometry of the straight line and plane - Exercise 10

 

11. Decide whether the given straight lines p and q are parallel or perpendicular to each other :
(work with normal vectors of the straight lines)

 

Analytic geometry of the straight line and plane - Exercise 11

 

12. Consider the two points A [3;2], B [–1;–1] and the vector a = (12;–5), where a = CB.


a) Find the coordinates of the point C.
b) Prove that the points A, B, C are vertices of a triangle.
c) Find the general equations of straight lines on which lie the sides of the triangle ABC.
d) Find the general equations of straight lines on which lie the medians of the triangle ABC.
e) Find the general equations of straight lines on which lie the altitudes of the triangle ABC.
f) Find the parametric equations of the straight line passing through the midpoints of the line segments AC and BC.
g) Find the slope-intercept equation of the straight line passing through the point A and parallel to the straight line BC.
h) Find the coordinates of the centroid T.
i) Find the perimeter of the triangle ABC.
j) Find the area of the triangle ABC.

 

13. Find the general equation of the perpendicular bisectors of line segments AB, AC and BC, if A [2;5], B [–3;9], C [6;12].

 

14. Prove that the points A [3;4], B [–1;2], C [1;3], D [–5;0] lie on one straight line. Find the parametric, the general and the slope-intercept equation of the straight line.

 

15. Find the parametric equations of a straight line passing through the point A [4;–1;9] which is parallel to

 

Analytic geometry of the straight line and plane - Exercise 15

 

16. Find the parametric equations of a straight line p passing through the point A [2;–1;2] perpendicularly to the plane π: xy + z + 13 = 0.

 

17. Find the parametric equations and the general equation of a plane ρ = ABC, A [–4;0;2], B [–2;1;1], C [1;–3;–2].

 

18. Find the general equation of a plane α which passes through the point A [2;1;4] and which is parallel to the plane β: x – 2y + 5z + d = 0.

 

19. Find the general equation of a plane σ which passes through the point A [1;2;0] and which is perpendicular to the straight line p: x = 3 – t; y = 4 + 2t; z = 1 – 2t; tR.

 

20. Consider a regular square pyramid ABCDV with vertices D [0;0;0], A [4;0;0], B [4;4;0], V [2;2;6]. Find the general equation of a plane BCV.

 

 

 

 

 

 

You might be also interested in:

- Vectors

- Relative Positions of Lines and Planes

- Conic Sections

- Matrices

 

 
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