7.2. Exercises on lines in space

Transcription

1 .. Exercises on lines in space Exercise : Change of support and direction vectors. Check whether P or Q are on g.. Use the point found in. to find both a new support vector and a new direction vector for g.. Assuming that g describes the motion of a particle, compare the speeds with the old and the new direction vector. a) P( ), Q( ) and g: x b) P( ), Q( ) and g: x Exercise : Intersects with the coordinate planes Find the intersects with the three coordinate planes E : x ; E : x and E : x. Use the intersects to draw a sketch of the line in a three dimensional coordinate system. Exercise : Line with given direction throu a given point Find the line throu the point P in the direction of b. Determine the speed of the particle. a) P( ) and b b) P( ) and b c) P( ) and b i j + k d) P( ) and b k Exercise : Line with given direction throu a given point a) Find the vector equation of the line h throu the point A( ) parallel to b i j + k. b) Calculate the magnitude of the direction vector. c) Find the coordinates of the point P on h with AP. Exercise : Line throu a two given points Find the vector equation of the line h throu the points P and Q: a) P( ) and Q( ) b) P( ) and Q( ) c) P( ) and Q( ) Exercise : Line throu a two given points a) Find the vector equation of the line h throu the points P( ) and Q( ). b) Find the coordinates of the point R on h with PR PQ. Exercise : Relations between lines. Find the intersects with the three coordinate planes E : x ; E : x and E : x.. Determine if the two lines g and h intersect and if they do, find the intersection point.. Find the angle between the two lines and decide whether they are parallel, perpendicular, skew or none of the above. e) g: x f) g: x g) g: x h) g: x

2 Exercise 8: Relations between lines Find all intersection points of the three lines, g, h and l: i ( j i ), h: x i + j ( i j ) and l: x i + j ( i j ). i + j ( i + j + k ), h: x j ( i j k ) and l: x i k ( j + k ). Exercise : Relations between lines in a parallelepiped The points P, Q, R and S are centroids of the side surfaces in the parallelepiped on the rit. Determine the relation of the lines g, h and i and find all intersection points. Exercise : Relations between lines in a prism The points E, F and G are centres of the edges in the parallelepiped on the rit. Determine the relation of the lines g, h and i and find all intersection points. Exercise : Motion on a strait line In this question the unit vectors i and j point due East and North, respectively. A port is located at the origin. One ship starts from the port and moves with velocity v ( i + j ) km h. a) Write down the position vector at time t hours. b) At the same time, a second ship starts 8 km north of the port and moves with velocity v ( i j ) km h.write down the position vector of the second ship at time t hours. c) Show that after half an hour the distance between the two ships is, km. d) Show that the ships meet and find the time when this happens. e) How long after the meeting are the ships 8 km apart? Exercise : Motion on a strait line At time t, two aircraft have position vectors j and k. The first moves with velocity i j + k and the second with velocity i + j k. a) Write down the position vector of the first aircraft at time t. b) Show that at time t the distance, d, between the two aircraft is given by d t 88t +. c) Show that the two aircraft will not collide. d) Find the minimum distance between the two aircraft. Exercise : Cartesian equation of a line a) Write down the Cartesian equation of the lines g and h in Exercise a) and b). x x x b) Write down the vector equation of the lines g: and h: x x, x. Exercise : Cartesian equation of a line. Determine whether the following pairs of lines are parallel, perpendicular, the same line or none of the above.. Find the intersection points of each pair. a) r c) e) x x + λ and r i k + μ( i + j + k ) b) r x and x x x and x λ +, x, x λ d) x t +, x t, x and r x x + and x x x e) r 8 + λ and x 8 x x x +

3 .. Solutions to the exercises on lines in space Exercise : Change of support and direction vectors a) P g because the equation PQ x r has no solution Q g because the equation OQ x has the solution r (Q is reached after r time units). With new supporting vector OQ and new direction vector we have g: x with double speed b) P g because the equation OP x has solution r. (Q is reached after r time units). Q g because the equation OQ x With new supporting vector OP same speed the original velocity. r and e.g. reversed direction vector. Because Q is reached after time unit, the vector Exercise : Intersects with the coordinate planes a) S ( ) for r, S ( ) for r and S ( ) for r b) S ( ) for r, S ( ) for r and S ( ) for r c) S ( ) for r, S ( ) for r and S ( ) for r d) S does not exist, S ( ) for r and S ( ) for r has no solution we have g: x with from starting point to Q is again Exercise : Line with given direction throu a given point Exercise : Line with given direction throu a given point b) speed is c) Since we need time units to reach P OP +. Exercise : Line throu two given points Exercise : Line throu two given points a) h: x b) OR +.

4 Exercise : Relation between lines a) g with S ( ), S ( ) and h with S ( ), S ( ), S ( ) are skew with α arccos, b) g with S ( ), S ( ) and h with S ( ), S ( ), S ( ) intersect in S( ) with r s, and α arccos, c) g with S ( ), S ( ) and h with S ( ), S ( ) are parallel. d) g with S ( ), S ( ), S ( ) and h with S ( ), S ( ), S ( ) intersect in S( ) with r resp s and α arccos, e) g with S ( ) S, S ( ) and h with S ( ) S, S ( ) are perpendicular and skew with α cos () f) g with S ( ), S ( ), S ( ) and h with S ( ), S ( ), S ( ) are parallel. g) g with S ( ), S ( ) S and h with S ( ), S ( ) and no S intersect in S( ) with r and s and α arccos, h) g with S ( ), S ( ), S ( ) and h with S ( ), S ( ), S ( ) are identical. (set r ) Exercise 8: Relations between lines a) g h {S } with position vector i + j, g i and h i {S } with position vector hi j b) g h {S } with i + j k, g i {S gi} with gi i + j, h i {S hi } with hi i + j k Exercise : relations between lines in parallelepiped g: x j + k ( i j k ), h: x i + j + k ( j k ) and i: x j ( i j + k ) g h {S } with i + j + k, i and g are skew and so are i and h. Exercise : relations between lines in prism g: x i ( i j k ), h: x j ( i + j k ) and i: x k ( i j + k ) g i {S gi } with gi i + j + k, h and g are skew and so are h and i. Exercise : Motion on a strait line a) first ship s : x t( i + j ) t b) second ship: s : x 8 j ( i j ) c) At time t the first ship is in, d) s s {S } with e) d 8 8 at t 8 and the second ship in (t t) (8 t t) t t.,, with distance d (,,) (, ),.

5 Exercise : Motion on a strait line a) first aircraft: a : x and second aircraft: a : x b) distance d(t) c) The equation d) d (t) (t) (t ) ( t) 88t 88 t 88t t (t t ) ( 8t t ) t t t / t / has no solution. t 88t. at t with sign change from to + rel Minimum at t with d() is also an absolute Minimum since d() and d for t. Exercise : Cartesian equation of a line x a) a) g: r x ; x and h: s x x + x b): g: r x x ; x and h: s x and h: x x Exercise : Cartesian equation of a line a) Intersection point S( ) at λ and μ b) skew and perpendicular c) Intersection point S( ) at μ and λ d) parallel d) Intersection point S(8 ) at μ and λ e) Intersection point S(8 ) at μ and λ