VECTOR NAME OF THE CHAPTER. By, Srinivasamurthy s.v. Lecturer in mathematics. K.P.C.L.P.U.College. jogfalls PART-B TWO MARKS QUESTIONS

Size: px

Start display at page:

Download "VECTOR NAME OF THE CHAPTER. By, Srinivasamurthy s.v. Lecturer in mathematics. K.P.C.L.P.U.College. jogfalls PART-B TWO MARKS QUESTIONS"

Dominic York
5 years ago
Views:

1 NAME OF THE CHAPTER VECTOR PART-A ONE MARKS PART-B TWO MARKS PART-C FIVE MARKS PART-D SIX OR FOUR MARKS PART-E TWO OR FOUR TOTAL MARKS ALLOTED APPROXIMATELY By, Srinivasamurthy s.v Lecturer in mathematics K.P.C.L.P.U.College jogfalls

2 VECTORS ONE MARK : 1.Define a vector and give an example. 2. Define a scalar and give an example. 3.Define a null vector ( or Zero vector ) 4. Define a unit vector. 5.Define co-initial vectors. 6.Define collinear vectors. 7. Define coplanar vector. 8. Define dierection cosines of a vector 9. Define dot product or scalar product of two vectors. 10.Define cross product or vector product of any two vectors. 11. If the position vectors of P & Q are 3i + 2j 7k and 4i + 7j 11k Then, Find PQ & PQ. 12. If a = 2i 3j + k, b = i + 2j k & c = 3i + 2j + 6k, then find 2a + b 3c. 13. Find the direction cosines of a vector 2i 3j + k 14.If the direction cosines of a vector are 1/4, 3/4 & n, then find n. 15. find the scalar product of the vectors 2i + 3j k & i - 2j 5k 16. If a = 2i j + 3k & b = i + 2j + k & c = 2i + j + k, find (a + b ) (b c ). 17.Prove that the vectors 3i j 2k & 2i -2j + 4k are orthogonal vectors.

3 18. Find m, if the vectors i + 3j 2k & 2i 4j + mk are orthogonal vectors. 19.Find the cosine of the angle between the vectors 3i + j 2k & 3i 5j 2k. 20. Find the angle between the vectors 2i + j + 2k & i 2j + 2k 21. Find the projection of the vector 2i + 3j 2k in the dierection of the vector i - 2j + 3k. 22. Find the angle between the vectors a + b & a - b if a = b. 23.If a + b = 5 and a is perpendicular to b, Find a - b. 24. If a & b are unit vectors and, a + b = 1, find a - b. 25. If a, b, c are 3 vectors, such that a + b + c = 0 and a = 1, b = 2 & c =3, find the value of a b + b c+ c a. 26.Find the cross product of the vectors j - 3k & i - j + 2k. 27. If b = 3a + c, prove that a x b = a x c. 28. Prove that (2a + b ) x ( a + 2b ) = 3 ( a x b ) 29.Show that the vectors 5i + 6j + 7k, 3i + 20j + 5k & 7i - 8j + 9k are coplanar. 30. If the vectors 2i - 3j + mk, 2i + j - k & 6i - j + 2k are coplanar,then, Find m. 31.Prove that [ i - j, j - k, k - i ] = 0

4 TWO MARKS 1.Prove that the position vector of a point dividing the points A & B internally in the ration m:n is given by r = mb + na m + n where a & b are the position vectors of A & B w.r.t some fixed point. 2. ABCD is a parallelogram and E is the point of intersection of two diagonlas, if O is any fixed point, prove that, OA + OB + OC + OD = 4 OE 3. If A = ( 2, 3, -4 ) and B = ( 1, ) Find the co-ordinates of the point dividing AB internally in the ratio 2 : Show that the vectors 2i - j + k, i - 3j - 5k & 3i - 4j - 4k form a right angled triangle. 5. Show that the points with position vectors i + 2j + 3k, - i -j + 8k & - 4i + 4j + 6k form an equilateral triangle..6. If a = 5i - j -3k, b = i + 3j + 5k, show that, ( a + b ) & ( a - b ) are orthogonal vectors. 7. Prove that, (i) a + b ² = a ² + b ² + 2 a b (ii) a - b ² = a ² + b ² - 2 a b 8. Prove that ( i ) a + b ² + a + b ² = 2 { a ² + b ² } ( i i ) a + b ² - a + b ² = 4 a b 9. If a = 3, b = 5 & c = 7 and a + b + c = 0, find the angle between the vectors a & b. 10. If a & b are unit vectors inclined at an angle of 60 to each other, find a + b.

5 11.If a & b are unit vectors inclined at an angle θ to each other, show that a b = 2 Sin(θ/2) 12. ABC is an equilateral triangle of side a then prove that, AB BC + BC CA + CA AB = - 3/2 a² 13.Find a unit vector perpendicular to both the vectors 2i 2j + k & 4i + j k. 14. Find a unit vector perpendicular to the plane determined by the points ( 1, -1, 2 ), ( 2, 0, -1 ) & ( 0, 2, 1 ). 15. Find the Sine of the angle between the vectors 4i + 3j + 2k & i j + 3k. 16. Find the area of the parallelogram whose adjecent sides are represented by the vectors i + j + k & i j + k. 17. Find the area of the parallelogram whose diagonals are represented by the vectors - 4 i +2 j + k & 3 i 2 j - k. 18. Find the area of the triangle,two of whose sides are represented by the vectors 3i + 4j & 5i + 7j + k. 19. Find the area of the triangle whose vertices are represented by the position vectors i+ 3j + 2k, 2i j + k & - i + 2j + 3k. 20.Find the perpendicular distance of A ( 1, 4, -2 ) from the line segment BC, where B ( 2, 1, -2 ) & C = ( 0, -5, 1 ). 21. Prove that a x ( b + c ) = prove that, a x b ² + a b ² = 2 { a ² b ² } 23. If a x b = 4 & a b = 2, Find a ² b ². 24. If θ be the angle between the vectors a & b, find the value of a x b a b 25. If a + b + c = 0, prove that a x b = b x c = c x a

6 26. Find the volume of the parallelepiped whose co-terminal edges are represented by the vectors 2i + j k, 3i 2j + 2k & i - 3j 3k. 27. Find the vector triple product a x ( b x c ), when a = 2i + 3j k, b = i + 2j 5k & c = 3i + 5j - k 28. Find the value of ( a x b ) x c, when a = ( 1, 2, 3 ), b = ( 2, 1, 2 ) & c = ( 3, 3, 2 ) 29. prove that, a x ( b x c ) = 0

7 VECTORS THREE MARKS 1. In a regular hexagon ABCDEF, Show that AB + AC + AD + AE + AF = 3 AD 2. Prove that position vector of the centroid of a triangle ABC is 1/3( a + b + c ), where, a, b & c are the position vectors of the vertices A, B & C w.r.t. some fixed point O. 3. If the position vectors of the points P and Q are 2 i + 3j + 4k and 3 i 2 j 3 k, find the direction cosines of the vector PQ and hence prove that, Cos²α + Cos²β + Cos²γ = 1 4.Show that the points ( 1, 2, 1 ), ( 2, 4, 2 ) ( 4, 3, -2 ) & ( 3, 1, -3 ) are the vertices of a parallelogram 5. Find a unit vector perpendicular to both the vectors a & b, Also, find the Sine of the angle between the vectors a & b, where, a = 6 i 2j + k & b = 3 i + j 2 k. 6. If a, b & c are the position vector of the vertices of a triangle ABC, Prove that,vector area of the triangle ABC = ½ ( a x b ) + ( b x c ) + ( c x a ) square units. 7. Prove that, [ a + b b + c c + a ] = 2 [ a b c ] 8.Find a unit vector which should lie on the plane determined by the vectors 2 i + j + k & i + 2 j + k and perpendicular to i + j + 2k. 9.Show that, i x ( a x i ) = 2 a 10.Show that the points ( - 6, 3, 2 ), ( 3, -2, 4 ), ( 5, 7, 3 ) & ( -13, 17, -1 ) are coplanar.

8 VECTORS 4 OR 5 MARKS 1. A, B, C & D are the points with position vectors 3 i 2 j k, 2 i + 3 j 4 k, - i + 2 j + 2 k & 4 i + 5 j + λk respectively. If the points A, B, C & D lie on a plane, Find the value of λ. 2.Find a unit vector which is coplanar with a & b and perpendicular to a, where, a = 2i + j + k & b = i + 2j k 3. If ( a x b ) x c = a x ( b x c ), then prove that either a is parallel to c or b is perpendicular to both a & c 5. Prove by vector method, that the medians of a triangle are concurrent. 6. Prove that diagonals of a parallelogram bisect each each other. 7. Prove by vector method that, The angle in a semi circle is a right angle. 8. In any triangle ABC, prove by vector method a b c ( a ) = = SinA SinB SinC ( b ) a² = b² + c² - 2bcCosA ( c ) a = bcosc + c CosB 9. Show that the points with position vectors, ( i ) i + j + k, 2i + 3 j + 4 k, 3 i + j + 2 k & - i + j ( ii ) - 6a + 3 b + 2 c, 3 a 2 b + 4 c, 5 a + 7 b + 3 c & - 13 a + 17 b c are coplanar.

9 VECTORS 6 MARKS 1. Define Dot product and vector product of any two co-initial vectors. If a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k, Prove that, a b = a 1 b 1 + a 2 b 2 + a 3 b 3 and i j k a x b = a 1 a 2 a 3 b 1 b 2 b 3 2. Prove that [ a b c ] = [ b c a ] = [ c a b ] & also show that [ a b b ] = 0 3. Prove that [ a x b, b x c, c x a ] = [ a, b, c ]² & Also, If a x b, b x c & c x a are coplanar, then prove that, a, b & c are coplan 4. prove that ( a x b ) x c = ( a c ) b - ( b c ) a 5. Prove that, Using vector method, Cos ( A - B ) = CosACosB + SinA SinB & Cos ( A - B ) = CosACosB + SinA SinB 6. prove that, Using vector method, Sin ( A B ) = SinACosB CosASinB and Sin ( A + B ) = SinACosB + CosASinB

CHAPTER 10 VECTORS POINTS TO REMEMBER

For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two