Vector, Exercícios de Matemática

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VECTOR

By:- Nishant Gupta

For any help contact:

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-

1. The necessary and sufficient condition for three points with position vectorsa, b,c

to be collinear is that

there exist scalars x, y, z not all zero such thatxa yb zc 0

   where x + y + z = 0.

2. If A and B are two points with position vectorsa

andb

respectively, then the position vector of a point

C dividing AB in the ratio m : n

(i) Internally is,

m n

mb na

(ii) Externally is,

m n

mb na

3. If S is any point in plane of (^) ABC, then SA SBSC 3 SG, where G is the centroid of ΔABC. 4. Ifa

andb

are two non zero vectors inclined at an angle θ, then

(i)a b |a||b|

(^)   cos θ (ii) Projection ofa

onb

= b

a

|b|

a b

(iii) Projection vector ofa

onb

= b

|b|

a b

b

|b |

a b

(iv) 2 (a b)

|b|

|a|

| a b|

(v)

|b|

(a b) (a b) |a|

     (vi) cos θ = 

|a||b |

a b

(vii)a  b|a||b|sin nˆ

, where n is a unit vector perpendicular to the plane ofa

andb

(ix) Unit vectors perpendicular to the plane ofa

andb

is ±

|a b|

a b

(x) Ifa

,b

are unit vectors at an angle θ, then sin |a b|

, cos |a b|

, tan

|a b|

|a b|

5. Area of ΔABC = |BC CA|

|BC BA|

|AB AC|

6. Ifa, b,c

are the PV the vertices A, B, C of ΔABC, then Area of ΔABC = |a b b c c a|

Length of the perpendicular from C on AB =

|a b|

| a b b c c a|

VECTOR

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-

 

2 2

2 2

2 a b

(a xb) a.b

is

(a) 1/2 (b) 3/

(c) 5/2 (d) 4/

2. The area of the triangle determined by the

vectors 3i + 4j and - 5i + 7j is

(a) 141 (b) 132

(c) 41 /2 (d) N/T

3. Points whose position vectors are

j

i 52

j,a

i 8

j, 40

i 3

60    are collinear if

(a) a = 40 (b) a = -

(c) a = 20 (d) N/T

4. Adjacent sides of ||gm are along

j. s

i

j&b 2

i 2

a     

between diagonals

(a) 30

o & 150

o (b) 45

o & 135

o

(c) 90

o & 90

o (d) N/T

5. a &b

are unit vectors such that a b

 3 is ┴

to 7 a b

 5 , then angle between a b

& is

(a) π/2 (b) π / 3

(c) π /4 (d) N/T

6. If the unit vectorsA

andB

are inclined at π

and |A

- B

|/2 is

(a) 0 (b) π/

(c) 1 (d) π/ 4

7. A particle acted upon by forces

  

3 i 2 j  5 k

and

  

2 i j  3 kis displaced form a point P to

a point Q whose respective position vectors

are

  

2 i j  3 k and

  

4 i 3 j  7 k. The work

done by the force is

(a) 77 units (b) 24 units

(c) 63 units (d) 48 units

8. A force F = 6i + λ j + 4k acting on a particle

displaces it from A (3,4,5)to B (1,1,1). If the

work done is 2 units, then λ is

(a) -10 (b) – 2

(c) 5 (d) 2.

9. Length of longer diagonal of llgm constructed

on 5 a 2 b

 &a 3 b

. Given| b| 3 &|a| 2 2

angle betweena &b

is π/

(a) 15 (b) √

(c) √593 (d) √

10. The vector k

j 3

i x

  is rotated through an

angle θ and doubled in magnitude, then it

becomes k

j 2

i ( 4 x 2 )

4   . The value of x is:

(a)  2 / 3 , 2  (b) 1 / 3 , 2 

(c)  2 / 3 , 0  (d) (2, 7)

11. If P and Q Be two given points on the curve y

= x + 1/x such that OP.I = 1 and OQ.I = -

where I is a unit vector along the x-axis, then

the length of vector 2OP + 3OQ is

(a) 5 5 (b) 3 5

(c) 2 5 (d) 5

ASSIGNMENT

VECTOR

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-

12. Let A, B, C be three vectors such that A (B + C)

• B. (C + A) + C. (A + B) = 0 And |A| = 1,

|B| = 4 , |C| = 8 ,then |A + B + C| equals

(a) 13 (b) 81

(c) 9 (d) 5

13. If the unit vectorsA

andB

are inclined at an

angle 2θ and |A

- B

|<l then for θ [0,], θ

may lie in the interval

(a) ( /6 , /3) (b) ( /6 , /2 ]

(c) ( 5/6 ,  ] (d) [/2 ,5/6 ]

14. If unit vectorsA

andB

such that STP [ A

B

A

xB

] = 1/4 then A

andB

are inclined

(a) π/6 (b) π/

(c) π/3 (d) π/ 4

15. If A

andB

unit vectors then greatest value

of |A

- B

| + |A

+B

| is

(a) 2 (b) 4

(c) 2√2 (d) √

16. Let a,b,c be three vectors such that | a | = | c

|= l; |b |=4 and | b x c | = 15. If b-2c = λa

Then a value of λ is

(a) 1 (b) - l

(c) 2 (d) - 4

17. Moment of couple formed by forces 5 k

i

-5 k

ˆ i

ˆ (^)  acting at ( 9 ,-1 , 2 ) & ( 3 , -2 , 1 )

(a) k

j 5

i

   (b) k

j 5

i

(c) k

j 10

i 2

2   (d) k

j 10

i 2

18. Vector r

which is equally inclined to co-

ordinate axes such that | r | = 15 3 is

(a) k

j

i

  (b) 15 k

j

i

(c) 7^ k

j

i

  (d) None

19. For 3 vectors u, v,w,

which of the following

expressions is to any of remaining three?

(a)u.( v w)

 (b)(v w).u

(c)v.( u w)

 (d) w u v

( x ).

20. If a  bc 0 ,|a| 3

, | b| 5 &|c| 7

, then

 θ betweena &b

is

(a) a = 40 (b) a = -

(c) a = 20 (d) N/T

21. If 2 out of 3 vectors a ,b,c

are unit vectors,

a  bc 0

& 2(a. b b.c c.a

  ) + 3 = 0, then

third vector is of length-

(a) 3 (b) 2

(c) 1 (d) N/T

22. Let a, b,c

be 3 vectors such that

a.( b c)b.(ca)c.(ab) 0

and

| a| 1 ,|b| 4 ,|c| 8

then| a b c|

  equals

(a) 13 (b) 81

(c) 9 (d) N/T

23. Leta b

(^)  is orthogonal tob

& a 2 b

(^)  is

orthogonal to a

, then

(a) |a

| = √2 |b

| (b) |a

| = 2 |b

(c) |a

| = |b

| (d) 2 |a

| = |b

24. Magnitude of projection of vector i 2 j k

on vector k

j 7

i 4

4   is

(a) 3 (b) 3 6

(c) 6 /3 (d) N/T

25. Position vectors of A & B are

k&

j

i 2

2   k

j 4

i 4

2  . Length of internal

bisector of BOA of  AOB is

(a)

(b)

(c)

(d) N/T

26. Magnitude of moment of force - k

j 8

i 6

acting at point k

j 3

i

2   about point

k

j

i 2

(a) 211 (b) 0

(c) 54 (d) N/T

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-

(c) 8 (d) 6

42. A vector a = ( α ,β, γ) makes an obtuse angle

with y- axis, equal angles with with b = ( β ,

• 2γ , 3 α ) & c = (2γ , 3α ,-β ) and is

perpendicular to d = (1, -1, 2). If |a | = 2√3 ,

then the vector a is

(a) (2, 2,-2) (b) (-2. -2, -2)

(c) (-2,-2, 2) (d) (2,-2,-2).

43. If a, b,c

are non coplanar vectors such that

[k( a b),k b,kc]

2

 =[a ,b c,c]

 , k has

(a) no value (b) exactly one value

(c) exactly two values

(d) exactly three values

44. Vector OP = i- 3j -2k and OQ = -3i + j -2k.

Then OM, the positive vector of bisector of

angle POQ, is

(a) i - j - k (b) 2 ( i + j – k )

(c) i + j + k (d) – i + j + k

45. Leta, b,c

be P.V. of vertices of a ∆ ABC

whose circumcenter is origin then

orthocenter is equals

(a)a b c

  (b) (a b c

(c) (a b c

(^)   ) /2 (d) N/T

46. Vectors a, b & c are related by a = 8b &

c = - 7c then angle between a & c is

(a) 0 (b) π /

(c) π/2 (d) π

ANSWER (VECTOR)

a c b c b c d a b b

d c c a c d b b c b

c c a d a b a a c a

c c a b d a d a b d

Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-

b d a b a d