MATHEMATICS

CLASS ROOM ASSIGNMENT MATHEMATICS

Batch: XIICVECTORSCra No: 100

Q3Let = 2+ 2 and = + . If is a vector such that . = ||, | | = 2and the angle between (x ) and is 30(, then |(x ) x | =

(a)

(b*)

(c) 2

(d) 3

Q4.If denote the vectors respectively, show that is parallel to to to respectively.

Q5.If are non coplanar unit vectors such that and are non parallel, then find the angles which makes with and .

Ans.( = , ( =

Q6.Let be a unit vector and be a non zero vector not parallel to . Find the angles of the triangle, two sides of which are represented by the vectors and .

Q7.Find the scalars ( and (, if . And where and are non collinear vectors and (, ( rescalars.

Ans. ( = 1, ( = 2n( + , n ( Z

Q8.Show that: .

Q9.If are vectors such that , prove that .

Q10.If are three non coplanar vectors and form a reciprocal system of vectors, then prove that

(i)

(ii)

(iii)

(iv) are non coplanar iff so are

Q11.If and be the reciprocal system of vectors, prove that

(i)

(ii)

Q12.If and are two sets of non coplanar vectors such that for i = 1, 2, 3, we have , then show that [] [] = 1.

Q13.If the vectors , , are not coplanar, then prove that the vector, ( x ) x ( x ) + (x ) x (x ) + (x ) x (x ) is parallel to

Q14.Let be three non coplanar unit vectors. The angle between be (, an angle between and be ( and between andbe (. If A(cos (), B() and C(), then show that

, where .

Q15.Find a vector in the plane and orthogonal to and with its projection along equal to .

Ans.

Q1Let =-, =, = and is a unit vector such that . = 0 = [

EMBED Equation.3

EMBED Equation.DSMT4 ]. Find Ans: (

Q2If and are any two orthogonal unit vector and is any vector . Evaluate (.) + (.) + .(x) (x)

Ans:

Q3The position vectors of the vertices A, B, C of a tetrahedron ABCD are + + , , 3 respectively. The altitude from the vertex D to the opposite face ABC meets the median line through A of triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is , find position vector E Ans: 3 or +3+3

Q4Let , , be non coplanar unit vectors, equally inclined to one another at an angle (. If x+ x = p+q+r, find scalars p, q, r in terms of (

Ans: p = r =

Q5For any two vectors and , prove that

(i) (.)2 + (x )2 = ||2 ||2

(ii) (1 + ||2) (1 + ||2) = (1 . )2 + (++ x )2Q6Let and be unit vectors. If is a vector such that + ( x ) = , then prove that | x . | ( and that equality holds if only if is perpendicular to

Q7.Let ABC and PQR be any two triangles in the same plane. Assume that the perpendicular from the points A, B, C to the sides QR, RP , PQ respectively are concurrent. Using vector methods, prove that the perpendiculars from P, Q, R to BC, CA, AB respectively are also concurrent

Q8.Show, by vector methods, that the angular bisectors of a triangle are conurrent and find and expression for the position vector of the point of concurrency in terms of the position vectors of the vertices. Ans:

Q9.If are three non coplanar vectors, prove that .Q10.If , are three non coplanar unit vectors and (, (, ( are the angles between and and and respectively and are unit vectors along the bisectors of the angles (, (, (, respectively. Prove that

.Q11.If = x1 + y1 + z1,= x2 + y2 + z2,= x3 + y3 + z3, prove that [

EMBED Equation.3

EMBED Equation.3 ] = = [

EMBED Equation.3

EMBED Equation.3 ]Q12.If , , are three given non coplanar vectors, show that any arbitrary vector in space is expressible in the form

=

EMBED Equation.3 +

EMBED Equation.3 +

EMBED Equation.3 , where (1 = , (2 = , (3= , ( =

Q13.Let V be the volume of the parallelepiped formed by the vectors = a1+ a2+ a3, = bi+ b2+ b3 and

= c1+ c2 + c3. If ar, br, cr, where r = 1, 2, 3 are non negative real numbers and ( ar + br + cr ) = 3L,

show that V ( L3Q14.A transversal cuts the sides OL, OM and diagonal ON of a parallelogram at A, B, C respectively. Prove that .

Q15.If O is the circumcentre and O( the orthocenter of a triangle ABC, prove that

(i) where S is any point in the plane of triangle ABC.

(ii)

(iii)

(iv)

where is a diameter of the circumcircle.

Q16.If , , be any three non coplanar vectors, then prove that the points

are coplanar if .

Q17.If any point O within or without a tetrahedron ABCD is joined to the vertices and AO, BO, CO, DO are produced to cut the planes of the opposite faces in P, Q, R, S respectively, then prove that

Q1.In a (OAB, E is the midpoint of OB, and D is a point on AB such that AD : DB = 2 : 1. If OD and AE interect at P, determine the ratio OP : OD using the vector method.

Ans: 3 : 5

Q2.In a triangle ABC, D and E are points on BC and AC respectively, such that BD = 2DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find using vector methods.

Ans:

Q3.Prove by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the midpoints of the parallel sides.

Q4.Let = + and = 2 . Show that the point of intersection of the lines and is 3 +

Q5.Using vectors, show that the internal bisector angle A of a triangle ABC divides the side BC in the ratio AB : AC

Q6.Show that the internal bisectors of the angles of a triangle are concurrent.CLASS ROOM ASSIGNMENT MATHEMATICS

Batch: XIIA, CVECTORSCra No: 102.2

Q1. Solve the vector equation , where are two given vectors

Ans:

Q2. If ( 0, find the vector which satisfies the equation

Ans:

Q3. Solve the vector equation given that and are given vectors such that . = 0

Ans: , where x ( R

Q4. Solve the vector equation , where , are two given vectors and k is a given scalar.

Ans:

Q5. Let and be the unit vector such that x = . Also, is any vector such that = 3, = 4 and = 2. Find in terms of and

Ans:

Q6. If , are three noncoplanar vectors, solve the vector equation

Ans:

Q7. Let be two given noncollinear unit vectors and be a vector such that . Then prove that |

Q8. Solve the vector equation for :

Ans: ,

Q9. Solve for :

Ans:

Q10. Solve for :

Ans: , where ( is an arbitrary constant

Q11. If are noncoplanar vectors and is any vector, show that

Q12. If , express in terms of

Ans: , ,

Q13. If , express in terms of

Ans: , ,

Q14. Find the vector = (a, y, z), which makes equal angles with the vectors = (y, 2z, 3x) and = (2z, 3x y) and is perpendicular = (1, 1, 2) with = 2 and the angle between and unit vector j is obtuse

Ans: (2, 2, 2)

Q15. Find the scalar ( and ( if are noncollinear

Ans: ( = , ( = 1

Q16. Show that the vector equations and , where are non coplanar, are consistent only if = 0 and solve for

Q17. Vectors each of magnitude makes equal angles 60( with each other. If = , = and = , express in terms of

Ans: , ,

Q18. Let be unit vectors such that , . Find in terms of

Ans:

PAGE Topic : Vectors

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