Vectors: Lines in Space A straight line can be determined by any two points in space. A line can also be determined by specifying a point on it and a direction. The direction would be a non-zero parallel vector. IB Math HL Notes 6B.4.1 The diagram to the right shows line L passing through point A and Parallel to vector v. Point M is on line L. For M to be on the line L, must be a multiple of v. i.e.,, where t is a scalar. Parametric Equations of a line The equation can be written in coordinate form: And for two vectors to be equal, their components have to be the same: and Which leads to x =, y = and z = The line that passes through the point A ( and parallel to the vector v = (a, b, c) has parametric equations:,, Example: Find the parametric equations of the line through A(1, -2, 3) and parallel to Example: Find parametric equations of the line through the points A(1, -2, 3) and B(2, 4, -2). Vector Equation of a line can also be expressed in terms of position vectors: is the position vector of the fixed point is the position vector of point The line L could be described by finding the difference of the two position vectors, Hence: Vector equation of a line Where r is the position vector of any point on the line and is the position vector of the fixed point (A). is the vector to the given line.
Example: Find a vector equation of the line that contains A(-1, 3, 0) and is parallel to Hint: IB Math HL Notes 6B.4.2 Rewrite the equation above by combining like components. Thus Practice: Find a vector equation of the line passing through A(2, 7) and B (6, 2) Practice: Find parametric equations for the line through A(-1, 1, 3) and parallel to the vector Now compare your parametric values with the points A, B and C by plugging in t = 0, 1, and 2 Line segments can be written using parametric equations (like above), but by limiting the parameter to the of the segment. LINE SEGMENTS Example: Parametrize the line segment between A(3,7,1) and B(1, 4, 2). Start by letting A be the fixed point and finding the direction vector = (,, ). Now find the parametric equations: Notice that when t = 0, the line starts at point. When t = 1, the line is at B(1, 4, 2). So, to get JUST THE LINE SEGMENT, we restrict the values of t to The final answer would be the parametric equations above combined with the restrictions on t to limit the line to the segment. OR the SEGMENT could be rewritten using a vector equation: Note when, and when. This traces the line segment from A to B. So in this example the vector equation of the segment would be Simplify
Practice: Parametrize the segment through A(2, -1, 5) and B(4, 3, 2). Hint: start with IB Math HL Notes 6B.4.3 Show that your components in the vector form have the same values as if you had used the parametric individual forms:,, with a, b and c coming from the components of =(,, ) Now use your equation Hint: find to find the midpoint of the segment. You can divide a line segment up into any ratio, i.e. by simply finding and plugging in the t ratio value. Symmetric (Cartesian) equations of lines Another set of equations for a line can be obtained by eliminating the parameter from the parametric equations. If, then we can rearrange to get to the Cartesian equations: Note: in formulae a, b, c are represented by l, m, n vector equation: is The fixed point A coordinates ( on line L appear in the numerators and the components a, b, and c of a direction vector appear in the denominators of these fractions. Example: Find the Cartesian equations of the line through A(3, -7, 4) and B(1, -4, -1). Find Cartesian equations again, but using B as the fixed point. Example: Let L be the line with Cartesian equations Find a set of parametric equations for L. Hint: Pick off the fixed point and the vector parallel to L. Find the vector equation r
IB Math HL Notes 6B.4.4 Intersecting, parallel and skew straight lines In a plane, lines either coincide ( line), intersect (one in common) or are parallel ( in common) Things are different in SPACE.. In addition to the three cases above, there is the case of the skew straight lines Skew straight lines are not and do not. Lines are parallel if the direction vectors are parallel. Parallel vectors are scalar of each other. Or you can find the angle between the two vectors. An angle of or would mean the lines are either coincident (same) or parallel. If the case for coincidence is there, you can check by examining a on one of the lines and seeing it if is also on the other line. Example: Show that the following two lines are parallel. (Be sure to check if the lines coincide) Hint: pick off the vectors parallel to each line. If the direction vectors are not parallel, the lines either or are. We check first to see if the lines intersect. If so, we find the coordinates of the point of intersections. Example: The lines have the following equations. Show that the lines are skew. Step 1: Determine if the lines are parallel Step 2: For the lines to intersect, there must be some point M( some values of t and s. Solve for the two variables using two equations which satisfies the equations of both lines for Step 3: and then plug into the 3 rd equation to see if it works. If it does, the system is consistent. If it is false, the system is inconsistent and the lines are SKEW.
IB Math HL Notes 6B.4.5 Example: The lines have the following equations. Determine if they are skew or intersecting. If intersecting, find the point of intersection. REDO example above in vector form: Rewrite and Then set the two equations equal to each other. The vector form of the equation of a line in space is more revealing when we think of the line as the path of an object, placed in an appropriate coordinate system and starting at position A( and moving in the direction of. Application of lines to motion Note the direction is a unit vector this makes it able to give direction without changing magnitude.
IB Math HL Notes 6B.4.6 Example: A model plane is to fly directly from a platform at a reference point (2,1,1) toward a point (5,5,6) at a speed of 60m/min. What is the position of the plane (to the nearest metre) after 10 minutes? Hint: find unit vector in the direction of the flight Equation: Example: An object is moving in the plane of an appropriately fitted coordinate system such that its position is given by, where t stands for time in hours after start and distances are measured in km. a) Find the initial position of the object b) Show the position of the object on a graph at start, 1 hour and 3 hours after start c) Find the velocity and speed of the object. Example: After leaving an intersection of roads located at 3 km east and 2 km north of a city, a car is moving towards a traffic light 7 km east and 5 km north of the city at a speed of 30 km/h. (Consider the city as the origin for an appropriate coordinate system) a) What is the velocity vector of the car? b) Write down the equation of the position of the car after t hours. c) When will the car reach the traffic light?