Vectors - Series 5 - The Line in 2 and 3 Dimensions Part 1: Vector Equation and Cartesian Equation Problem 1: (Repetition) Find a 2D-Cartesian-equation of the following lines: a) line passing through A(−1/1) and B(5/3), b) line passing through A(5/ − 2) with zero x = 1. c) line parallel to the y-axis passing through the point A(−3/ − 7). d) line parallel to the x-axis passing through the point A(8/ − 5). e) Which line among the ones from a)-d) is not the graph of a linear function? Problem 2: (Repetition) h is parallel to the line g connecting the points A(4/−1) and B(−2/−5.5) and passes through the point C(2/6.5). Find the equation of the the center parallel (Mittelparallele) m of g and h. Problem 3: (Repetition) In which point and at which angle do the lines g and h intersect? a) g : y = 4x − 7, h : y = 7x − 4 c) g : y = 23 x − 1, h : y = 23 x + 1 b) g : y = −x − 3, h : y = 2x + 6 d) g : y = −2x + 3, h : y = 21 x + 6 Problem 4: Find appropriate vectors to show that the angle between two lines with slopes m1 and m2 resepctively is given by the formula cos φ = p 1 + m 1 m2 p . 1 + m21 1 + m22 Can you recognize the orthogonalit-condition for the slopes m1 and m2 of two lines in the 2Dplane? Problem 5: Find three different vector equations for the lines passing through P and Q: a) P (2/ − 1/4), Q(−2/ − 3/6) b) P (−1/4/ − 14), Q(5/1/1) c) P (p1 /p2 /p3 ), Q(q1 /q2 /q3 ) 0 6 2 −2 4 , ~y = ~x = w ~= ~v = Problem 6: ~u = −6 0 −7 3 2 are the direction vectors of various lines. Which are the corresponding slopes of these lines? Problem 7: Find point and angle of intersection for the lines g and h. Test orthogonality before you calculate the angle of intersection. 3 5 2 −12 a) g : ~r = +λ , h : ~r = +λ 1 12 5 5 0 3 9 −1 b) g : ~r = +λ , h : ~r = +λ −3 5 4 1 5 3 −1 −6 c) g : ~r = +λ , h : ~r = +λ 8 −2 4 4 Problem 8: Reflect the point P in the line g and find the mirror image P 0 of P . 3 4 −7 2 a) g : ~r = +λ , P (12/ − 3) b) g : ~r = +λ , P (0/ − 4) −1 1 0 −3 Problem 9: Let the triangle 4ABC with vertices A(2/1), B(5/5), C(−3/11) be given. The point S is the point of intersection of the median sa and the height hc . The point T is the point of intersection of the bisector ωβ and the side b. Find the distance between S and T . Hint: First find vector equations for each of the four relevant lines such that the components of the direction vectors are as small in absolute value as possible. Part 2: Basic Geometric Problems Problem 10: (Point on Line-Test) Test whether the points P (5/ − 9/ − 4) and Q(3/ − 5/ − 2) lie on the line −2 1 g : ~r = 1 + λ 6 ? 4 2 Give a step-by-step instructions how to solve this problem in general. Problem 11: (Special Lines) How can you see from the vector equation of a line that it is a) parallel to the x-axis (y-axis, z-axis)? b) parallel to the xy-plane (yz-plane, xz-plane)? Problem 12: (Trace Ponts) The trace points of a line are the intersection points of a line with the coordinate-planes. We denote the trace points with the xy-plane, the yz-plane and the xz-plane in this order Sxy , Syz and Sxz . Draw a sketch of a line in 3D space and indicate its trace points. Find the trace points of the line 1 −2 g : ~r = 1 + λ 6 , 2 4 and formulate step-by-step instructions for the general case. Problem 13: (Relationships between Lines) a) Consider two lines with vector equations g1 : ~r = a~1 + λb~1 and g2 : ~r = a~2 + λb~2 . In 3D-space there are 4 possible spatial relationships which describe the mutual position of the lines g1 and g2 . Try to find all 4 possible spatial relationships. b) Draw a structured chart (a tree-diagram) showing the mathematical criteria to be tested in order to decide the spatial relationship between two lines in 3D-space. c) Consider all 6 pairs of two lines which can be formed from the set of the following four lines g1 , g2 , g3 und g4 and classify each of them according to the spatial relationship of the two lines 1 3 g1 : ~r = 1 + λ −1 , 3 2 2 0 g2 : ~r = 2 + λ −1 , −1 7 −2 1 g3 : ~r = −2 + λ 1 , 1 5 6 3 g4 : ~r = −3 + λ −3 . −3 4 Problem 14: (Angle of Intersection) The lines g and h intersect. Find the angle of intersection. 4 −2 3 0 a) g : ~r = 0 + λ 2 , h : ~r = −2 + λ 3 1 5 3 1 b) Formulate step-by-step instruction for the general case of the problem in a). c) Find the equation of the bisector ωα in the triangle 4ABC with vertices A(1/2/−1),B(3/3/1) and C(−2/2/3). Problem 15: (Distance of a point from a line) We try to find the distance of the point P (−1/4/ − 14) from the line g with equation: −1 2 g : ~r = 1 + λ −1 . 1 −2 a) What does this distance mean geometrically? Draw a sketch. b) Why does the foot Q of the perpendicular dropped from P to the line g have coordinates of the following form Q(−1 + 2λ/1 − λ/1 − 2λ)? c) Use the dot product to find an equation for λ, and hence find Q. d) Hence what’s the distance d(P, g) from P to g? e) Formulate step-by-step instructions how to find the distance of a point P from a line g. f ) The point P is to be reflected in the line g. Complete your sketch from a), and find the mirror image P 0 . Problem 16: (Distance between skew lines) The following two lines g and h are skew: 2 2 2 2 g : ~r = 7 + λ 3 , h : ~r = −3 + µ 0 . −6 0 7 −1 a) What’s the meaning of the distance between two skw lines? By which geometrical property −−→ can you charcterise the vector P Q which connects the two foots P and Q of this distance? b) Use the vector equations of the lines g and h to express the coordinates of P and Q, and −−→ find the component representation of P Q in terms of the parameters λ and µ. c) Translate the conditions of orthogonality into parametric equations and solve them. d) Find the foots P and Q and also the distance d(g, h). e) A plane A flies from O(0/0/0) towards the point P(5000/2000/800) whereas plane B flies from Q(−2500/ − 3000/900) towards the point R(4000/3000/600). How close do the two flight paths get? Part 3: Further Problems to Practise Problem 17: Find a vector equation of a line a) passing through A(−4/0/3) and parallel to the z-axis. b) passing though A(7/5/3) and parallel to the y-axis. c) passing through the midpoint of A(−2/1/5) and B(4/2/ − 5) and parallel to the x-axis. Problem 18: Do the points A(5/4/2), B(0/ − 11/ − 7) and C(7.5/11.5/7) lie on the straight line with vector equation 1 2 g : ~r = −5 + λ 3 ? 2 −4 Problem 19: Which special property does the line g exhibit? 0 0 a) g : ~r = 0 + λ 0 b) g : ~r = 1 7 −2 2 c) g : ~r = 5 + λ 5 d) g : ~r = 0 0 4 −1 0 + λ 0 3 0 0 3 1 + λ 3 5 4 Problem 20: Let the triangle 4ABC with vertices A(1/10/2), B(−1/3/1), C(−2/0/6) be given. Find the trace points of its sides with the xy-plane and show that the three points are collinear. Problem 21: Divide the segment with endpoints A(−4/5/ − 2) and B(5/ − 1/4) into three parts of equal length. Find the coordinates of the endpoints of these parts. Problem 22: Which is the relationship between the lines g and h? If they intersect, also find the angle of intersection. a) 2 −1 −3 3 g : ~r = −1 + λ 2 , h : ~r = 5 + µ −2 1 1 4 1 b) 3 0.8 2 −4 g : ~r = 2 + λ 0.2 , h : ~r = 6 + µ −1 −1 −1 1 5 c) 3 1 −0.6 −2 g : ~r = 1 + λ −1 , h : ~r = 6 + µ 5 −1 2 0.2 3 d) 4 −2 3 0 g : ~r = 0 + λ 2 , h : ~r = −2 + µ 3 1 5 3 1 e) 0 5 0 7 g : ~r = −3 + λ 6 , h : ~r = 2 + µ −3 0.5 3 −1 0 f) −3 −1 4 1 g : ~r = 6 + λ 2 , h : ~r = 0 + µ 0 0 1 −3 0 g) −3 8 −16 3 g : ~r = 5 + λ −5 , h : ~r = −6 + µ 3 −2 −1 9 9 h) 2 14 −7 6 g : ~r = −10 + λ −6 , h : ~r = −4 + µ 6 3 3 6 24 Problem 23: Find vector equations for (both!) bisectors of the lines g and h: −4 3 2 3 g : ~r = 2 + λ 0 , h : ~r = 2 + µ 1 8 1 2 1 Problem 24: The follwng cube has side-length 6 and the point C is a midpoint of an edge. Find the distance of the point A from the line given by the points C and D. Also find the mirror image B 0 if the point B is reflected in the line passing through C and D. D 1 0 0 1 1 0 B 1 C0 1 0 0 1 A Problem 25: Find the distance between the following skew lines g and h. a) −1 0 3 6 g : ~r = 0 + λ 1 , h : ~r = 2 + µ −1 3 1 1 2 b) 4 4 2 5 g : ~r = 1 + λ 2 , h : ~r = 3 + µ 2 −3 −1 3 −2
