MISCELLANEOUS QUESTIONS

CHAPTER 5 : VECTORS SM025

MISCELLANEOUS QUESTIONS

1. Given the point P(−4, 2, − 3) , the straight line L : x + 2 = y = z + 1 and the plane
4 −3 5

 : 2x + y + 2z = 9 . Find
a) an acute angle between the straight line L and the plane  .
b) an intersection point between the straight line L and the plane  .
c) a Cartesian equation of the plane containing the point P and the straight line L .

2. P , Q and R are three points in space where PQ = a and PR = b . Given
a = 2i + 2 j −k
b =i + 2j + 2k

a) Find the area of the triangle PQR .
b) Find the parametric equations of the line L passing through the point R(2, 0, 3) and

parallel to vector a .
c) If u = ( b a + a b) and v = ( a b − b a) , evaluate u  v . Hence, interpret

the geometrical relationship between u and v .

3 a) If u and v are nonzero vectors, show that u  (u  v) = 0 .
b) Find a unit vector perpendicular to u = −2i + 3 j − 3k and v = 2i − k .

4. The plane 1 contains a line L with vector equation r = t j and a point P(3, − 1, 2) .
a) Find a Cartesian equation of 1 .
b) Given a second plane  2 with equation x + 2y + 3z = 4 , calculate the angle between
1 and  2 .

5. Given nonzero vectors p and q are perpendicular. Prove that

a) p+q 2 = p 2+ 2

q.

b) p + q = p − q .

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CHAPTER 5 : VECTORS SM025

6. Given the points A(1, 2, − 2) , B(2, 4, 6) and C(−4, 3, − 1) . Find the area of the triangle
ABC .

7. Given two planes 1 : x + 2y + z = 1,  2 : 2x − y + 4z = 1 and the straight line
L : x − 2 = y + 3 = z −1.
2 45
a) Find an acute angle between the planes 1 and  2 .
b) Write the equation of L in parametric form. Hence, find the intersection point between
the straight line L and the plane  2 .
c) Find a Cartesian equation of the plane which is orthogonal to the straight line L and
passes through the point (1, 2, − 3) .

8. Given two straight lines,

L1 : x −1 = y+2 = z and L2 : x+2 = y = z−4.
−3 8 −3 10 10 −7

a) Show that L1 and L2 are not parallel and find the acute angle between the two

straight lines.

b) Determine intersection point between L1 and plane  : 2x − y + 5z + 25 = 0 .

c) Find an equation of the plane containing L1 and L2 .

9. Given P , Q and R are three points in a space where
PQ = a = 3i − j + k , PR = b = 2i + j − 3k

and the coordinates of R is (3, 0, 1) .
a) Hence, show that

i) a and b are not perpendicular.
ii) ab 2 = a 2 b 2 − ( ab )2 .
b) Find the area of triangle PQR .
c) Find the Cartesian equation for the
i) plane that passes through the points P , Q and R .
ii) line that passes through the point R and perpendicular to the plane in part (i).

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CHAPTER 5 : VECTORS SM025

10 a) If p = 3i − j + 2k and q = 2i + 2 j − k , show that
pq 2 = p 2 q 2 − ( pq )2 .

b) Given a triangle ABC with AB = 2a and AC = 3b . Use the result in part (a), show
that the area of the triangle is 3 a 2 b 2 − (a  b)2 .
Hence, deduce the area of the triangle if a = p and b = q .

11. Given a line l : x = 2 − t , y = −3 + 4t , z = −5 − 3t and two planes
1 : 2x − y + 7z = 53 and  2 : 3x + y + z = 1. Find

a) the point of intersection between the line l and the plane 1 .
b) the acute angle between the line l and the plane 1 .
c) the acute angle between planes 1 and  2 .

12. Find the angle between the line l :  x, y, z  =  1, 3, − 1  + t  2, 1, 0  and the
plane  : 3x − 2y + z = 5 .

13. Given four points A = (−2, − 8, 4) , B = (2, − , − 1) , C = (0, − 9, 0) and
D = (−4, − 3, 7) . Determine the value of  if AB  (AC  AD) = 64 .

14 a) If the line l1 :  x, y, z  =  1, 1, 2  + t  2, − 1, 3  does not intersect with the
plane 1 : Ax + By + Cz = 0 , show that 2A − B + 3C = 0 .
Hence, find the equation of plane 1 if the plane passes through the point (1, 0, 1) .

b) Given the line l2 : x = x0 + tv1 , y = y0 + tv2 , z = z0 + tv3 , the plane
 2 : x − y + 2z = 0 and a point (x0 , y0 , z0 )  (0, 0, 0) is on the plane.
i) If l2 is perpendicular to the plane  2 , show that
 v1, v2 , v3  = v2  −1, 1, − 2  ; v2  0 .
ii) Give one example of the equation of straight line which satisfy part (b)(i).

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CHAPTER 5 : VECTORS SM025

15. Given vectors p = 3i − 6 j +  k and q = i − 4 j + 5k where  and  are constants.
a) Find the values of  and  if p and q are parallel.
b) Given  = 1, find  if p and q are perpendicular.

16. Given three points P(−3, 2, −1) , Q(−2, 4, 5) and R(1, − 2, 4) . Calculate the area of
triangle PQR .

17. Given the line L: x −1 = y −3 = z − 2 and the planes 1 : 2x − y − 2z = 17 and
2 −1 −3

 2 : − 4x − 3y + 5z = 10 .

Find

a) the intersection point between L and 1 .
b) the acute angle between 1 and  2 .
c) the parametric equations of the line that passes through the point (2, −1, 3) and

perpendicular to the plane  2 .

18. The line L1 and L2 passes through the point R(2, 4, − 3) and S(8, − 5, 9) in the
direction of 2i − 3 j + 4k and i − 2 j + 3k , respectively.

a) State the equations for lines L1 and L2 in the vector form. Hence, calculate the acute
angle between the lines L1 and L2 .

b) Find the equation of plane containing the line L1 and the point (7, − 3, 5) in the
Cartesian form.

c) Determine whether the line L2 is parallel to the plane x + 5y + 3z = 5 .

19. The points A(2,3,5) , B(4, − 2,3) and C(−2,0,7) lie on the plane  . A straight line
L passes through the points P(2,8,9) and Q(5,6, − 4) .

a) Find AB , AC and AB  AC . Hence, obtain the Cartesian equation of the plane  .
b) Find the vector equation of the straight line L .
c) Find the acute angle between the straight line L and plane  .

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CHAPTER 5 : VECTORS SM025

20. Point R(−1, 2,3) and Q(x, y, z) are such that PQ and RS are normal to the planes
M and N . Planes M and N are parallel. Points S(−2, 4,6) and P(1,3, 2) are on
the planes M and N respectively.
a) State PQ .
b) Find
i) vector equation of the line that passes through point P and parallel to RS .
ii) the Cartesian equation of the plane N .
iii) determine the acute angle between the intersection of plane N and another plane
with equation 2x − 3y + 2z = 1.

c) Describe PQ = 2RS . Then, find point Q .

ANSWERS

1 a)  = 450 b) (2, − 3, 4) c) 2x + y − z = −3

2 a) 4.03 unit2 b) x = 2 + 2t , y = 2t , z = 3 − t

c) u  v = 0 ; u and v are perpendicular

^ 1 (−3i − 8 j − 6k)

3 b) n =
− 109

4 a) 2x − 3z = 0 b) 58.70
6) 21.4 unit2

7 a)  = 69.10 b) x = 2 + 2t , y = −3 + 4t , z = 1+ 5t ; 1, − 5, − 3 
 2

c) 2x + 4y + 5z = −5

8 a)  = 60.20 b) (−2, 6, − 3) c) − 26x − 51y −110z = 76

9 b) 6.12 unit2 c i) 2x +11y + 5z = 11 ii) x − 3 = y = z −1
b)  = 46.10 2 11 5
10 b) 33.1 unit2
11 a) (5, − 15, 4) c)  = 60.50

12)  = 28.60
13)  = −10

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CHAPTER 5 : VECTORS SM025

14 a) x − y − z = 0 b ii) x = −1 + t , y = 1 − t , z = 1 + 2t

15 a)  = 15 ,  = 2 b)  = − 29
2 3

16) 20.4 unit 2

17 a) (5, 1, − 4) b)  = 450

c) x = 2 − 4t , y = −1 − 3t , z = 3 + 5t

18 a) L1 : r1 = 2i + 4 j − 3k + t (2i − 3 j + 4k) , L2 : r2 = 8i − 5 j + 9k + s (i − 2 j + 3k) ;
 = 6.980

b) 4x + 4y + z = 21

c) parallel

19 a) AB = 2i − 5 j − 2k , AC = −4i − 3 j + 2k , AB  AC = −16i + 4 j − 26k ;

− 8x + 2y −13z = −75

b) r = (2i + 8 j + 9k) + t(3i − 2 j − 13k)

c) 42.80
20 a) (x −1)i + (y − 3) j + (z − 2)k

b i) r = i + 3 j + 2k + t(−i + 2 j + 3k) ii) −x + 2y + 3z =11 iii)  = 82.60

c) PQ is same direction with RS and magnitude PQ is two times magnitude RS ;
Q(−1, 7, 8)

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