Lines & Planes 8 K/U 13 Comm 10 Think Unit 6 Test 12 App Name ________________________________ Knowledge/Understanding (8 marks) Identify the choice that best completes the statement or answers the question. ____ 1. a. b. A line passes through the points A(2, 3) and B(-2, 1). Find a vector equation of the line. c. [x, y] = [-4, -2] + t[2, 3] d. [x, y] = [2, 3] + t[-4, -2] ___ 2. A line has slope -3 and y-intercept 5. Find a vector equation of the line. a. [x, y] = [0, 5] + t[1, -3] c. [x, y] = [1, -3] + t[0, 5] b. [x, y] = [5, 0] + t[0, -3] d. [x, y] = [-3, 5] + t[-3, -3] ____ 3. a. b. Write the scalar equation of the plane with normal vector c. d. and passing through the point (3, 2, 1). ____ 4. A plane passes through the origin and has the direction vectors [1, 2, 3] and [-1, 3, -2]. Find a scalar equation of the plane. a. c. b. d. ____ 5. The parametric equations of a plane are a. b. ____ 6. . Find a scalar equation of the plane. c. d. Find the intersection point of the two lines: and . c. (1, -1) d. a. b. and [x, y, z] = [-5, 4, -5] ____ 7. In three-space, find the intersection point of the two lines: + t[3, -1, 4]. a. (-5, 4, -5) c. b. d. (3, 2, 1) ____ 8. In three-space, find the intersection point of the two lines: a. (-7, -7, -3) b. (1, -3, -3) c. (2, -1, 0) d. (2, 1, -3) and . Communication (13 marks) True/False Indicate whether the statement is true or false. ____ 9. A normal vector to a line is perpendicular to that line. ____ 10. The direction vector of a line is an indicator of its slope. ____ 11. There is no scalar equation of a line in three space. ____ 12. A plane written in scalar form cannot be written in vector form. ____ 13. Lines intersect in a point in two-space but do not intersect in three-space. ____ 14. Any three non-collinear points in space will define a unique plane. ____ 15. In three-space there are four possibilities for the intersection of two lines. ____ 16. Skew lines are lines that are not parallel but lie in parallel planes. ____ 17. A line will always intersect with a plane. Completion Complete each statement using any of the words below: WORD BANK: [ point, line, plane, skew, intersection, collinear, normal, scalar, vector] 18. A vector perpendicular to a line is called a _______________ vector. 19. If the dot product of the direction vector of a line and the normal of a plane is not equal to zero, then the intersection of the line and the plane is a _______________. 20. Two planes are parallel if their normals are _______________. 21. If the normals of two planes are not collinear, then the planes will always intersect in a _______________. Thinking (10 marks) 22. Find a normal vector of the plane . [2] 23. Find the intercepts of the plane [x, y, z] = [5, 4, 3] + s[1, 0, 1] + t[-1, -4, -2] . [2] 24. The point [2] 25. By analysing the normals, determine if the two planes intersect in a line, are parallel and distinct, or are coincident. [2] 26. By analysing the normals, determine if the three planes intersect in a point. lies on the plane . Find the value of k. [2] Application (12 marks) 27. In 3-space, find the distance between the skew lines 28. Determine the intersection point of the line and [4] and the plane [4] 29. Determine the solution to the following system: [4]