Cartesian vector and parametric equations of lines IB
Cartesian, vector and parametric equations of lines IB HL Section 11. 3
Notation of lines in space Vector equation Cartesian equation Rearrange the first parametric equation to make t the subject. coordinate on the line the vector part Do this for each of the other two parametric equations. Parametric equations This can be made up from the vector equation: Make up the equations for y and z. The cartesian equation becomes:
Notation of lines in space 1. A line is parallel to the vector 3 i- j+ 2 k and passes through the point (3, -2, 4). Write the vector, parametric and Cartesian equation of the line. Vector: Parametric: X=3+3 t Y=-2 -t Z=4+2 t Vector:
Vector equation of a line 1 a) Find the plane’s coordinate after 1 hour. (9, 3) b) Find the plane’s coordinate after 2 hours. (14, 5) c) Find the plane’s coordinate after t hours. Vector equation Parametric equation 4 (4, 1) 2 5 10 A coordinate on the line. The vector part of the line.
Vector equation of a line 2 Find the vector and parametric equations of the straight line that passes through A(x 1, y 1) and B(x 2, y 2). Find the vector that connects A to B. Vector equation will be: Parametric equation will be: or Find the vector and parametric equations of the straight line that passes through A and B that have position vectors and. Vector equation: Or using the B coordinate and the vector BA. or Parametric equation: or
Intersecting lines Vector lines can intersect, Write down two vector line although they do not have to. equations. Example 2 ships plan to meet at a buoy (B). Ship 1 starts at (2, 3) and moves along the vector. Ship 2 starts B has coordinates (x, y). at (13, 10) and moves along the vector. B S 2 Solving this gives t=3, u=2. Check this out with both lines gives the coordinates: B (5, 12) S 1
Questions Two ants set off to meet each other at point M. The first ant starts at (7, 1) and the second ant starts at (18, 13). The ants are moving along the vectors, a) Find the coordinates of M. M (12, 11) b) Find the distance that the first ant covers. c) Find the distance that the second ant covers.
Vectors in space Two vectors in space will either be parallel, intersecting or skew. Parallel lines Intersecting lines Skew lines Lines in space can be parallel with an angle between them of 0 o. Lines that meet in space will intersect each other. The angle at the point of intersection is Lines that do not meet each other in space, but are not parallel, are called skew. These lines can still have an angle between them.
Parallel 3 d lines The parametric equations of two lines in space are given as: Show that the two lines are parallel. Turn the parametric equations into vector equations: Take the vector parts of these equations and note that one is a multiple of the other: b is -3 a therefore the lines are parallel.
Intersecting 3 d lines Show that the following 2 straight lines intersect and find the angle between them. If the lines intersect each othere must be coordinate (x, y, z) at the point of intersection. So the task is to find values of s and t that satisfy all the equations. Set up two simultaneous equations: Solving these equations gives: s=2 and t=-1 This equation works for the x and y parts of the lines. Check it also works for the z part. The coordinate at the point of intersection is (4, 5, -2).
Skew 3 d lines Show that the following 2 straight lines are skew and find the angle made between them. As before, set up two simultaneous equations and solve them. from the x and y parts of the line. Solving these will give t=5 and u=-7. Now check this works for the z part of the line: Not equal. As the simultaneous equations do not satisfy all 3 coordinates there is no point of intersection and therefore the lines are skew.
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