Vectors2 planes

Vectors (Planes) Level 1 1. The position vectors of points A and B are given by OA = + + i j k °°°± , 2 3 4 OB = + + i j k °°°± . i) Find a vector equation of the line passing through A and B . ii) Find the position vector of the point where this line meets the plane 2 1 8 1     ⋅ − =       r . Ans: i) 1 1 1 2 , 1 3 λ λ         = +             ∈ r ℝ , ii) 3 5 7           2. [ACJC/07/1/13] The planes and 2 Π have equations 1 1 2 0 0 3 1 3 0 α β −             = + +             −       r and 3 2 6 1 r     ⋅ =     −   respectively, where , α β ∈ R . a) Show that 1 Π and 2 Π are parallel. [2] b) The line 1 l which passes through the point A , with position vector, i k − , and is parallel to 3 2 7 i j k − + meets 2 Π at B . Find i) the position vector of B ; [4] ii) the length of projection of AB in the plane 2 Π . [3] Ans: bi) 2 2 8 −       −   , bii) 12 21 7 3. The position vectors of points A and B are given by 2 3 OA = + − i j k °°°± and 5 3 OB = + j k °°°± respectively. Given that C is a point on the line AB such that OC °°°± is perpendicular to the line AB . i) Find the position vector C . ii) Find a vector perpendicular to the plane containing O , A and B . Ans: i) 11/ 6 19 / 6 2 / 3         −   , ii) 14 6 10     −       1 Π

4. [VJC/10/2/3] The plane ∏ and the line l have equations 1 . 1 4 a     =       r and 1 1 2 0 2 1 λ         = +             r respectively, where λ is a real parameter and a is a constant. a) It is given that 1 a = − . Find i) the acute angle between ∏ and the plane 0, z = [2] ii) the exact perpendicular distance of the point (1, 3, 2) from ∏ . [3] b) It is given that 3. a = − Find the acute angle between l and . ∏ [3] Ans: ai) 0 54.7 , aii) 2 3 3 , b) 0 25.2 5. Find the vector equation of the plane determined by the points ( ) 0,1,1 , ( ) 2,1 3 − , ( ) 1,3,2 and also find the coordinates of the point in which this plane intersects the line ( ) ( ) ( ) 1 2 2 3 3 5 λ λ λ = + + − + − − r i j k Ans: 4 3 1 2     ⋅ − = −       r , ( ) 3 1, 8 − − 6. The planes 1 π and 2 π have equations 0 1 2 2     ⋅ =       r and 2 1 12 4     ⋅ − =       r respectively. The point A has coordinates ( ) 7,2,0 and O is the origin. i) Verify that A lies in both 1 π and 2 π . ii) The planes 1 π and 2 π meet in the line L . Show that a vector equation of the line L is 7 3 2 2 , 0 1 λ λ         = +         −   ∈   r ℝ iii) The point B has coordinates ( ) 7, 4,9 − and the foot of the perpendicular from B to 2 π is N . Find the coordinates of the point N . iv) Find the Cartesian equation of the plane 3 π containing the line L and is perpendicular to the plane 2 π . Ans: iii) ( ) 3, 2,1 − , iv) 2 3 x y z − + + = −

10. [IJC/07/1/10] The plane Π has equation 1 2 5 3     ⋅ =     −   r . The line l passes through the point P with position vector 7 4 6         −   and is parallel to 7 3 5         −   . i) State the perpendicular distance from the origin to the plane Π . [1] ii) Find the acute angle between the line l and the plane Π . [4] iii) Find the position vector of the foot of perpendicular from the point P to the plane Π . Hence find the position vector of the reflection of the point P in Π . [6] Ans: i) 5 14 14 , ii) 55.2 ° , iii) 5 0 0           , 3 4 6     −       11. [CJC/07/2/3] The equation of the plane 1 Π is given by x + 2 y − 2 z = 18. The position vectors of the points A and D are given by 2 i j − + and 2 5 3 i j k + − respectively. The foot of the perpendicular from A to the plane 1 Π is B . C is the point on BA produced such that BA : AC = 2:1. Find i) the position vector of B ; [3] ii) the perpendicular distance from A to the plane 1 Π ; [1] iii) the position vector of the point C ; [2] iv) the equation of the plane DBC in the form r n p ⋅ = ; [3] v) the length of projection of the line BD on 1 Π . [3] Ans: i) 0 5 4         −   , ii) 6, iii) 3 1 2 −     −       , iv) 2 5 9 4 −     ⋅ =       r , v) 5

12. The vector equation of line l which passes through the point A with position vector 5 2 3 + − i j k is ( ) 5 2 3 2 3 , α α = + − + − − ∈ 7i j r k i j k ℝ . The vector equation plane π which contains point B with position vector 5 2 2 − + + i j k is ( ) ( ) , 5 2 , 2 7 2 3 9 β γ β γ + ∈ = − + + + − − − + r i j k i j k i j k ℝ . i) Find the vector equation of plane π in scalar product form. ii) Find the position vector of point P on the line segment AB such that : 4:1 AP PB = . iii) The plane 1 π contains the line l and the point P . Find the vector equation of the plane 1 π in scalar product form. iv) Find the acute angle between plane π and 1 π . v) Find the line of intersection between π and 1 π . vi) Find the foot of the perpendicular from A to plane π . vii) Hence, find a) the perpendicular distance from A to plane π . b) the length of projection of AB on the plane π . Ans: i) 5 34 21 11 −     ⋅ − = −       r , ii) 3 2 1 −           , iii) 2 1 0 4     ⋅ =       r , iv) 2 π , v) 1/ 3 7 / 3 2 / 3 2 / 3 , 0 1 µ µ − −         = +             ∈ r ℝ , v) 285 / 62 23 / 31 131/ 62     −     −   , via) 5 62 1302 , vib) 7225 62 13. [ACJC/10/2/5] The plane p 1 has equation 2 3 x y z + − = . The line l 1 has equation 1 1 2 4 x z y + − = = . i) Show that the line 1 l is parallel to, but not contained in the plane p 1 . [2] ii) Find the cartesian equation of the plane p 2 which contains the line l 1 and is perpendicular to the plane p 1 . [3] iii) Find, in scalar product form, the vector equation of the plane p 3 which contains the point ( ) 4,1, 1 − and is perpendicular to both p 1 and p 2 . [2] Another line l 2 which is parallel to the vector 2 0 3         −   intersects the line l 1 at the point ( ) 1,0,1 − A . iv) Given that the line l 2 meets the plane p 1 at the point B , find the coordinates of B . [4] v) Find the sine of the acute angle between the line l 2 and the plane p 1 , and hence, find the length of the projection of the line segment AB on the plane p 1 , giving your answer in surd form. [4] Ans: ii) 3 2 4 x y z − + + = , iii) 2 r 1 5 4     ⋅ =       , iv) ( ) 1,0, 2 − , v) 5 78 , 1 318 6

16. [NYJC/08/1/11] Two planes 1 Π and 2 Π have equations ( ) ( ) ( ) 3 2 2 3 1 λ µ λ µ µ = + − + − + + − r i j k and 1 5 a b     − =       r i respectively. i) Show that the Cartesian equation of the plane 1 Π is 3 5 6 x y z + − = . [2] ii) Given that the point A (3, 2, 1) lies on 2 Π and that the two planes are perpendicular to each other, find the values of a and of b . [3] iii) Find a vector equation of l , the line of intersection between the planes 1 Π and 2 Π . [2] iv) B is a point with position vector – 6 i + 3 j + 11 k . Find the coordinates of the point C on 2 Π such that ∠ ACB = 90 ° . [3] v) The line L is the reflection of the line AB in the plane 2 Π . Find an equation for the line L and determine the acute angle between the lines l and L . [4] Ans: ii) a = 2, b = 1 (iii) 3 4 2 13 1 5 λ         = +             r , λ ∈ ℝ, iv) 3 25 3, , 2 2   −     , v) 3 3 2 2 , 1 13 α α         = +         −     ∈ r ℝ ; 82.1 0 .

17. [JJC/08/2/4] The position vectors of the points A , B and C are given as 1 3 0 2 , 0 , 1 3 1 2             −                   respectively. The point D lies on AB produced is such that AB:AD = 2:3. The equation of the plane 1 Π is 5 2 14. x y z − + = i) Find the position vector of the point D . ii) Find the vector equation of the plane 2 Π , in the form p = • n r , that contains the points A , B and C . iii) The equation of the line of intersection, L , of the two planes 1 Π and 2 Π , is given as 2 : , 7 r L p s q λ λ         = + ∈         −     r ℝ . Find the values of p , q , r and s . iv) Find the equation of the plane which is perpendicular to both the planes 1 Π and 2 Π and passing through the point B . Ans: i) 4 1 0           , ii) 1 1 5 2     ⋅ =       r , iii) 1, 2, 5, 9 p q r s = − = = = , iv) 5 9 8 7     ⋅ =     −   r . Level 2 18. [JJC/10/1/7] The line l passes through the points A and B with coordinates ( ) 1, 2, 3 − and ( ) 5, 14, 11 respectively. The plane p has equation 2 3 6 7. x y z + − = − i) Show that the line l is parallel but not contained on the plane p . [3] ii) Find the distance of the line l from the plane p . [3] iii) Find a cartesian equation of the plane which contains l and is perpendicular to p . [2] Ans: ii) 1, iii) 48 26 3 91 x y z − + − = 19. The plane Π passes through the point ( ) 2,0,0 A and is parallel to the vectors 2 − + + i j k and 3 2 − + i j k . i) Find the equation of Π in scalar product in scalar product form and verify that point ( ) 6, 5,1 B − also lies on Π . ii) Find the position vector of the foot of the perpendicular from the point ( ) 2,1 3 C − − to the plane Π . Hence, find the equation of 1 Π , the plane parallel to Π such that C is equidistant from planes Π and 1 Π . Ans: i) 1 1 2 1     ⋅ =       r , ii) 0 3 1         −   , 1 1 10 1     ⋅ = −       r

22. Given that the position vectors of the points A , B and C are 2 , 2 , 3 3 + − + − i k i j k i j respectively. Find the equation of the plane ABC in the form . p = r n . [3] If 1 l is the perpendicular bisector of the line segment BC on the plane ABC , explain why the equation of 1 l can be expressed as ( ) 2 , 2 2 λ λ = − + + + − ∈ r i i j k j k ℝ [3] The common line of intersection between plane ABC and 2 ∏ is 1 l . Given that 2 ∏ is perpendicular to plane ABC , find the vector equation of plane 2 ∏ . [2] Another plane 3 ∏ is perpendicular to both ABC and 2 ∏ and the three planes intersect at the point with position vector 2 2 − + i j k . Find the vector equation of 3 ∏ . [2] Ans: 1 0 3 1     ⋅ =       r , 1 1 3 1 −     ⋅ = −       r , 1 2 3 1     ⋅ = −     −   r 23. [AJC/05/FM/1/11] (modified) The planes 1 ∏ and 2 ∏ , which meet in a line l , have vector equations ( ) ( ) 1 1 2 3 4 3 5 α β = + + + − + + − r i j k i k i j k ( ) ( ) 2 2 2 3 3 4 2 α β = + + + + + + − r i j k i j i j k Respectively. The point B with position vector 2 3 + + i j k lies on both planes. i) Show that l is parallel to vector 4 − + i j k . ii) Write down a vector equation for l . iii) The point A on 1 ∏ has position vector 5 2 2 + + i j k . Show that AB is perpendicular to the line l . Hence find the distance of point A from the line l . iv) Explain why the angle between the vectors 4 − i k and + i j would give the angle between the planes 1 ∏ and 2 ∏ . Hence find the exact value of the cosine of the angle between 1 ∏ and 2 ∏ . v) Hence, or otherwise, find the distance of point A from the plane 2 ∏ Ans: ii) 1 1 : 2 1 , 3 4 l λ λ         = + −            ∈  r ℝ , iii) 2 17 units , iv) 2 34 17 , v) 2 3 units

24. [MJC/07/2/3] A line l has equation 4 3 0 2 . 5 1 λ λ −         = + −             ∈ r ℝ . i) Find the position vector of P , the foot of the perpendicular from the origin O to l . [3] ii) Find a cartesian equation of the plane 1 Π containing O and l . [3] iii) It is given that l also lies in a plane 2 Π with equation 2 3 1 k     ⋅ = −       r , k ∈ ℝ . Show that 7 2 k = . [2] iv) Find the angle between the planes 1 Π and 2 Π , giving your answer in degrees. [3] v) A third plane 3 Π has cartesian equation 2 7 x y z + + = . Determine the nature of the intersection of the three planes 1 Π , 2 Π and 3 Π . [3] Ans: i) 5 1 2 2 11 −     −       , ii) 10 19 8 0 x y z + + = , iv) 0 6.8 , v) no common intersections. 25. [VJC/10/1/8] The planes 1 ∏ and 2 ∏ have equations ( ) . 6 + − = r i j k and ( ) . 2 4 12 − + = − r i j k respectively. The point A has position vector 9 7 5 − + i j k . i) Find the position vector of the foot of perpendicular from A to 2 ∏ . [3] ii) Find a vector equation of the line of intersection of 1 ∏ and 2 ∏ . [2] The plane 3 ∏ has equation ( ) ( ) ( ) 3 3 9 b λ µ = + + + + + + − + + r i j k i j k i j k , where , λ µ are real parameters and b is a constant. Given that 1 2 3 , and ∏ ∏ ∏ have no point in common, find the value of b . [3] 3 ∏ meets 1 ∏ and 2 ∏ in lines 1 l and 2 l respectively. Without finding the equations of 1 l and 2 l , describe the relationship between 1 l and 2 l , giving a reason. [2] Ans: i) 3 5 2           , ii) 2 1 4 1 , 0 2 λ λ         = +             ∈ r ℝ , 3 b =

27. [NJC/07/1/12] The plane 1 Π has equation 1 2 4 1     ⋅ − =       r . The line l passes through two points A and B whose position vectors are 2 0 2           and 7 18 1 −         −   respectively. a) Show that the position vector of the point N on 1 Π such that BN is perpendicular to 1 Π is 1 2 7           . Hence find the perpendicular distance from B to 1 Π . [5] b) Verify that point A lies on 1 Π . Hence show that the reflection of l in 1 Π is parallel to 7 14 13 −         −   . [3] c) The plane 2 Π is perpendicular to 1 Π and contains l . Find the equation of 2 Π in the form p ⋅ = r n . [3] d) 3 Π is perpendicular to both 1 Π and 2 Π and contains the line BN . By considering the triangle ABN , or otherwise, determine the distance of A from 3 Π and the acute angle between l and 3 Π . [4] Ans: a) 8 6 , c) 4 2 8 0     ⋅ =       r , d) 30 , 0 15.6

28. [MJC/10/1/9] A line l passes through the points A and B with coordinates ( ) 0, 1,2 − and ( ) 1,0,1 respectively. i) Find the angle between OA and the line l . [2] ii) Hence, find the shortest distance from the origin to the line l . [1] A plane 1 π has equation ( ) 2 3 4. + + = r i j k i iii) Show that the line l lies on the plane 1 . π [2] A second plane 2 π contains the line l and is perpendicular to the plane 1 . π iv) Find a vector equation of 2 . π [2] A third plane 3 π is perpendicular to both the planes 1 π and 2 , π and is at a perpendicular distance of 3 units from the origin. v) Find possible vector equations of 3 π . [3] Ans: i) 0 39.2 , ii) 1.41, iv) 5 4 6 1     ⋅ − =       r , v) 1 1 1 3 or 1 3 1 1         ⋅ = − ⋅ =         − −     r r 29. [NYJC/10/2/4] a) The line l passes through the point A with coordinates (1,-1,1) and is parallel to the vector 2 2 + − i j k . The plane p has equation 1 1 3 1     ⋅ =       r . Find, in exact form, i) the position vector of B , the point of intersection between l and p , [3] ii) the sine of the acute angle between l and p , [2] iii) the shortest distance from A to p , [1] iv) the length of the projection of AB onto p . [2] b) Given that the system of linear equations 3 where , 3 4 9 3 y z y x y x x z α β α β + = + ∈ + + = + = ℝ has infinite solutions, obtain the numerical values of α and β . [4] Ans: ai) 5 1 3         −   , ii) 3 9 , iii) 2 3 3 iv) 26 2 3 , v) 1 1 1 3 or 1 3 1 1         ⋅ = − ⋅ =         − −     r r , b) 1 , 3 2 α β = − =

32. [NJC/08/1/10] l is given by the equation r = 2 i + k + λ ( i + 2 j – 3 k ), λ ∈ ℝ and the point P has position vector 2 i − j – 3 k relative to the origin. N is a point on l such that the line PN is perpendicular to l . i) Show that ( ) 1 5 17 13 7 = + + °°°± PN i j k . [2] ii) The plane 1 Π has equation ( ) 3 7 ⋅ + = r i k . Verify that l lies in 1 Π and find the perpendicular distance from point P to 1 Π . [4] iii) A second plane 2 Π contains l and P . Using your answers above, or otherwise, determine the acute angle between 1 Π and 2 Π . [2] iv) A third plane 3 Π has equation ( ) 5 22 ⋅ − + = r j k . Determine the position vector of the point of intersection between planes 1 Π , 2 Π and 3 Π . [2] Ans: ii) 4 10 , iii) 23.8 ° , iv) 1 2 4     −       33. [NJC/10/1/9] Relative to the origin O , the point A has position vector given by 0 1 0 OA     =       °°°± and A lies on the plane 1 Π with equation defined by r • 1 3 3 2     =       . Another plane 2 Π has equation y x = . The planes 1 Π and 2 Π intersect at line l . i) Find the vector equation of the line l . [1] ii) Show that the cosine of the acute angle between the planes 1 Π and 2 Π is 7 7 . [2] iii) Find the position vector of the foot of perpendicular, OF °°°± , from point A to the line l . Hence, find the exact length of projection of AF °°°± onto the plane 2 Π . [5] iv) Another plane 3 Π has equation 1 px qy + = , where p and q are real constants. Find the condition in which p and q must satisfy such that the planes 1 Π , 2 Π and 3 Π intersect at exactly one point. [2] Ans: i) 0.75 0.5 0.75 0.5 , 0 1 λ λ −         = + − ∈             r R , iii) 2 / 3 2 / 3 1/ 6 OF     =       °°°± , 3 6 , iv) p q ≠

34. [HCI/08/1/12] Referring to the origin O , two planes 1 Π and 2 Π are given by 1 1 : 2 13 4 Π     ⋅ =     −   r and 2 1 : 3 8 3 Π     ⋅ = −       r . i) Given that a point A (1, 7, − 10) lies on 2 Π , show that the perpendicular distance from A to 1 Π is 2 21 . [2] ii) Hence or otherwise find OB °°°± where B is the image of A when reflected in the plane 1 Π . [2] iii) Write down the Cartesian equations of both 1 Π and 2 Π . [1] Find a vector equation of the line of intersection of 1 Π and 2 Π . [1] iv) Find a vector equation of the plane which is the image of 2 Π when 2 Π is reflected in 1 Π . [3] Ans: ii) 3 1 6 −     −       , iii) 2 4 13 x y z + − = , 3 3 8 x y z + + = − , 55 18 21 7 , 0 1 λ λ         = − + −         ∈     r ℝ , iv) 3 18 29 1 7 10 , , 6 1 3 s t s t             = − + − + −             −       ∈ r ℝ Level 3 35. The line l and the plane P have equations ( ) , : l λ λ + = ∈ u v r ℝ ( ) : 0 P ⋅ = × r u v i) Show that l is contained in P . ii) For the case 1 2 1 −     =       u and 3 3 2     = −       v , find an equation of the line contained in P that is perpendicular to l and passes through the origin Ans: ii) 12 27 , 17 µ µ −     ∈       = r ℝ

38. [FM/99/1/11] Show that for any vectors u and v , ( ) ( ) 0 ⋅ = × × = ⋅ u u v v u v [1] With respect to the origin O , the points A , B , C have position vectors , , a b c respectively, and are such that , , , O A B C are non-coplanar. The mid-point of AC is M and the plane AOB is denoted by Π . The line l passes through M is perpendicular to Π . Show that the equation of l is ( ) 1 1 2 2 t = + + × r a c a b , where t is a parameter. [3] It is given that 1 5 , , 2, 12 2 2 ⋅ = ⋅ = − ⋅ = ⋅ = a a a c b b b c , and that a and b are perpendicular vectors. The line l meets Π at the point D . By expressing the equation of Π in a form involving two parameters, find the position vector of D in terms of a and b . [6] The line segment MD meets the plane OBC at the point H . Find the ratio : MH HD . [4] Ans: 2 3 = − + d a b , : 1: 4 MH HD =