1. The line L passes through the points A (3, 0, -1) and B (5, -1, 4). a) Find the vector equation of the line L. b) Determine whether or not the line L intersects the line with the equations (Marks available: 6) Answer Answer outline and marking scheme for question: 1 Give yourself marks for mentioning any of the points below: a) The equation for a line should be expressed as: r = a + λ b Where a is a point on the line, b is a vector parallel to the line and λ is any number. a = the first point A. b = point B minus point A. Putting these values into the equation of the line above, gives: (2 marks) b) Consider the point where the x values are the same for both lines, therefore: 3 + 2λ = 5 - 4μ Consider the point where the y values are the same for both lines, therefore: 0 - 1λ = 1 + 1μ Solving these equation simultaneously, gives: λ= -3, μ = 2. Putting these values into equation for line L, gives: z = -1 + 5λ = -16 Putting these values into equation for line r, gives: z = 11 + 3μ= 17. As the value of z is not the same, both the line cannot be at the same point in space (i. e. they do not intersect). (4 marks) (Marks available: 6) 2. A body of mass 0.5 kg moves so that its velocity at time t seconds is Find the magnitude of the momentum when t = 0 and t = 2. (Marks available: 3) Answer Answer outline and marking scheme for question: 2 Give yourself marks for mentioning any of the points below: At t = 0, the vectors equals The magnitude of the velocity equals Therefore the momentum at t = 0 equals 0.5 x 12.17 = 6.08 kgms-1. Performing the same calculation at t = 2, gives the momentum equal to 0.5 x 4.47 = 2.23 kgms-1. (Marks available: 3 marks) 3. Two lines A and B, have the following formulas: and a) determine whether these two lines intersect b) find the angle between them. (Marks available: 6) Answer Answer outline and marking scheme for question: 3 Give yourself marks for mentioning any of the points below: a) Matching the x-values gives: 4 - 4λ = 6 +2μ Matching the y-values gives: 0 + 8λ = -10 - 6μ Matching the z-values gives: -2 -2λ = -10 -2μ Solving the first two simultaneous equations gives: λ = 1, μ = -3. These values work in the third equation therefore the lines meet. Substituting λ = 1, μ = -3 into the equation of lines gives the point of intersection as being: x = 0, y = 4, z = -2. Therefore the lines meet at (0, 4, -2) (3 marks) b) The angle between the lines is the angle between the direction vectors, so using the scalar product we get, = = -0.855 This gives θ = 148.8o (or 180o - 148.8o = 31.2o). (3 marks) (Marks available: 6)