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:
Transcription
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
More information[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989]
VECTOR ALGEBRA o. Let a i be a vector which makes an angle of 0 with a unit vector b. Then the unit vector ( a b) is [MP PET 99]. The perimeter of the triangle whose vertices have the position vectors
More informationQuantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors.
Vectors summary Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. AB is the position vector of B relative to A and is the vector
More informationCreated by T. Madas VECTOR PRACTICE Part B Created by T. Madas
VECTOR PRACTICE Part B THE CROSS PRODUCT Question 1 Find in each of the following cases a) a = 2i + 5j + k and b = 3i j b) a = i + 2j + k and b = 3i j k c) a = 3i j 2k and b = i + 3j + k d) a = 7i + j
More informationPrepared by: M. S. KumarSwamy, TGT(Maths) Page
Prepared by: M S KumarSwamy, TGT(Maths) Page - 119 - CHAPTER 10: VECTOR ALGEBRA QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 06 marks Vector The line l to the line segment AB, then a
More informationthe coordinates of C (3) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4)
. The line l has equation, 2 4 3 2 + = λ r where λ is a scalar parameter. The line l 2 has equation, 2 0 5 3 9 0 + = µ r where μ is a scalar parameter. Given that l and l 2 meet at the point C, find the
More information1. The unit vector perpendicular to both the lines. Ans:, (2)
1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a
More informationFIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, TECHNICAL MATHEMATICS- I (Common Except DCP and CABM)
TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, 2010 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)
More informationVECTORS ADDITIONS OF VECTORS
VECTORS 1. ADDITION OF VECTORS. SCALAR PRODUCT OF VECTORS 3. VECTOR PRODUCT OF VECTORS 4. SCALAR TRIPLE PRODUCT 5. VECTOR TRIPLE PRODUCT 6. PRODUCT OF FOUR VECTORS ADDITIONS OF VECTORS Definitions and
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More informationPart (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
More informationChapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in
Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.
More informationVector Algebra. a) only x b)only y c)neither x nor y d) both x and y. 5)Two adjacent sides of a parallelogram PQRS are given by: PQ =2i +j +11k &
Vector Algebra 1)If [a хb b хc c хa] = λ[a b c ] then λ= a) b)0 c)1 d)-1 )If a and b are two unit vectors then the vector (a +b )х(a хb ) is parallel to the vector: a) a -b b)a +b c) a -b d)a +b )Let a
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationCOORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE. To find the length of a line segment joining two points A(x 1, y 1 ) and B(x 2, y 2 ), use
COORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE I. Length of a Line Segment: The distance between two points A ( x1, 1 ) B ( x, ) is given b A B = ( x x1) ( 1) To find the length of a line segment joining
More informationMAC Module 5 Vectors in 2-Space and 3-Space II
MAC 2103 Module 5 Vectors in 2-Space and 3-Space II 1 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in R 3. 2. Determine a scalar triple
More informationChapter 8 Vectors and Scalars
Chapter 8 193 Vectors and Scalars Chapter 8 Vectors and Scalars 8.1 Introduction: In this chapter we shall use the ideas of the plane to develop a new mathematical concept, vector. If you have studied
More informationVectors. Teaching Learning Point. Ç, where OP. l m n
Vectors 9 Teaching Learning Point l A quantity that has magnitude as well as direction is called is called a vector. l A directed line segment represents a vector and is denoted y AB Å or a Æ. l Position
More informationVECTORS. Vectors OPTIONAL - I Vectors and three dimensional Geometry
Vectors OPTIONAL - I 32 VECTORS In day to day life situations, we deal with physical quantities such as distance, speed, temperature, volume etc. These quantities are sufficient to describe change of position,
More information3D GEOMETRY. 3D-Geometry. If α, β, γ are angle made by a line with positive directions of x, y and z. axes respectively show that = 2.
D GEOMETRY ) If α β γ are angle made by a line with positive directions of x y and z axes respectively show that i) sin α + sin β + sin γ ii) cos α + cos β + cos γ + 0 Solution:- i) are angle made by a
More information( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.
Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line
More informationQ1. If (1, 2) lies on the circle. x 2 + y 2 + 2gx + 2fy + c = 0. which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c =
Q1. If (1, 2) lies on the circle x 2 + y 2 + 2gx + 2fy + c = 0 which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c = a) 11 b) -13 c) 24 d) 100 Solution: Any circle concentric with x 2 +
More informationThe Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /
The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will
More informationRegent College. Maths Department. Core Mathematics 4. Vectors
Regent College Maths Department Core Mathematics 4 Vectors Page 1 Vectors By the end of this unit you should be able to find: a unit vector in the direction of a. the distance between two points (x 1,
More informationPart r A A A 1 Mark Part r B B B 2 Marks Mark P t ar t t C C 5 Mar M ks Part r E 4 Marks Mark Tot To a t l
Part Part P t Part Part Total A B C E 1 Mark 2 Marks 5 Marks M k 4 Marks CIRCLES 12 Marks approximately Definition ; A circle is defined as the locus of a point which moves such that its distance from
More informationDefinition: A vector is a directed line segment which represents a displacement from one point P to another point Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have
More informationPROBLEMS 07 - VECTORS Page 1. Solve all problems vectorially: ( 1 ) Obtain the unit vectors perpendicular to each of , 29 , 29
PROBLEMS 07 VECTORS Page Solve all problems vectorially: ( ) Obtain the unit vectors perpendicular to each x = ( ) d y = ( 0 ). ± 9 9 9 ( ) If α is the gle between two unit vectors a d b then prove th
More informationCreated by T. Madas 2D VECTORS. Created by T. Madas
2D VECTORS Question 1 (**) Relative to a fixed origin O, the point A has coordinates ( 2, 3). The point B is such so that AB = 3i 7j, where i and j are mutually perpendicular unit vectors lying on the
More informationCHAPTER TWO. 2.1 Vectors as ordered pairs and triples. The most common set of basic vectors in 3-space is i,j,k. where
40 CHAPTER TWO.1 Vectors as ordered pairs and triples. The most common set of basic vectors in 3-space is i,j,k where i represents a vector of magnitude 1 in the x direction j represents a vector of magnitude
More informationSOLVED PROBLEMS. 1. The angle between two lines whose direction cosines are given by the equation l + m + n = 0, l 2 + m 2 + n 2 = 0 is
SOLVED PROBLEMS OBJECTIVE 1. The angle between two lines whose direction cosines are given by the equation l + m + n = 0, l 2 + m 2 + n 2 = 0 is (A) π/3 (B) 2π/3 (C) π/4 (D) None of these hb : Eliminating
More informationLecture 2: Vector-Vector Operations
Lecture 2: Vector-Vector Operations Vector-Vector Operations Addition of two vectors Geometric representation of addition and subtraction of vectors Vectors and points Dot product of two vectors Geometric
More informationGeometry. Class Examples (July 1) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 1) Paul Yiu Department of Mathematics Florida tlantic University c b a Summer 2014 1 Example 1(a). Given a triangle, the intersection P of the perpendicular bisector of and
More informationWorksheet A VECTORS 1 G H I D E F A B C
Worksheet A G H I D E F A B C The diagram shows three sets of equally-spaced parallel lines. Given that AC = p that AD = q, express the following vectors in terms of p q. a CA b AG c AB d DF e HE f AF
More informationSTRAIGHT LINES EXERCISE - 3
STRAIGHT LINES EXERCISE - 3 Q. D C (3,4) E A(, ) Mid point of A, C is B 3 E, Point D rotation of point C(3, 4) by angle 90 o about E. 3 o 3 3 i4 cis90 i 5i 3 i i 5 i 5 D, point E mid point of B & D. So
More information( ) = ( ) ( ) = ( ) = + = = = ( ) Therefore: , where t. Note: If we start with the condition BM = tab, we will have BM = ( x + 2, y + 3, z 5)
Chapter Exercise a) AB OB OA ( xb xa, yb ya, zb za),,, 0, b) AB OB OA ( xb xa, yb ya, zb za) ( ), ( ),, 0, c) AB OB OA x x, y y, z z (, ( ), ) (,, ) ( ) B A B A B A ( ) d) AB OB OA ( xb xa, yb ya, zb za)
More informationQ.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R
More informationDownloaded from
Triangles 1.In ABC right angled at C, AD is median. Then AB 2 = AC 2 - AD 2 AD 2 - AC 2 3AC 2-4AD 2 (D) 4AD 2-3AC 2 2.Which of the following statement is true? Any two right triangles are similar
More informationWhat you will learn today
What you will learn today The Dot Product Equations of Vectors and the Geometry of Space 1/29 Direction angles and Direction cosines Projections Definitions: 1. a : a 1, a 2, a 3, b : b 1, b 2, b 3, a
More informationContact hour per week: 04 Contact hour per Semester: 64 ALGEBRA 1 DETERMINANTS 2 2 MATRICES 4 3 BINOMIAL THEOREM 3 4 LOGARITHMS 2 5 VECTOR ALGEBRA 6
BOARD OF TECHNICAL EXAMINATION KARNATAKA SUBJECT: APPLIED MATHEMATICS I For I- semester DIPLOMA COURSES OF ALL BRANCHES Contact hour per week: 04 Contact hour per Semester: 64 UNIT NO. CHAPTER TITLE CONTACT
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationThe Cross Product. In this section, we will learn about: Cross products of vectors and their applications.
The Cross Product In this section, we will learn about: Cross products of vectors and their applications. THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is a
More informationUNIT 1 VECTORS INTRODUCTION 1.1 OBJECTIVES. Stucture
UNIT 1 VECTORS 1 Stucture 1.0 Introduction 1.1 Objectives 1.2 Vectors and Scalars 1.3 Components of a Vector 1.4 Section Formula 1.5 nswers to Check Your Progress 1.6 Summary 1.0 INTRODUCTION In this unit,
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationBasic Mathematics - XII (Mgmt.) SET 1
Basic Mathematics - XII (Mgmt.) SET Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Model Candidates are required to give their answers in their own words as far as practicable. The figures
More information8. Find r a! r b. a) r a = [3, 2, 7], r b = [ 1, 4, 5] b) r a = [ 5, 6, 7], r b = [2, 7, 4]
Chapter 8 Prerequisite Skills BLM 8-1.. Linear Relations 1. Make a table of values and graph each linear function a) y = 2x b) y = x + 5 c) 2x + 6y = 12 d) x + 7y = 21 2. Find the x- and y-intercepts of
More informationVECTORS IN COMPONENT FORM
VECTORS IN COMPONENT FORM In Cartesian coordinates any D vector a can be written as a = a x i + a y j + a z k a x a y a x a y a z a z where i, j and k are unit vectors in x, y and z directions. i = j =
More informationChapter 2. The laws of sines and cosines. 2.1 The law of sines. Theorem 2.1 (The law of sines). Let R denote the circumradius of a triangle ABC.
hapter 2 The laws of sines and cosines 2.1 The law of sines Theorem 2.1 (The law of sines). Let R denote the circumradius of a triangle. 2R = a sin α = b sin β = c sin γ. α O O α as Since the area of a
More informationPOINT. Preface. The concept of Point is very important for the study of coordinate
POINT Preface The concept of Point is ver important for the stud of coordinate geometr. This chapter deals with various forms of representing a Point and several associated properties. The concept of coordinates
More informationVectors Practice [296 marks]
Vectors Practice [96 marks] The diagram shows quadrilateral ABCD with vertices A(, ), B(, 5), C(5, ) and D(4, ) a 4 Show that AC = ( ) Find BD (iii) Show that AC is perpendicular to BD The line (AC) has
More informationPREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared by IITians.
www. Class XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) ALP ADVANCED LEVEL PROBLEMS Straight Lines 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared b IITians.
More informationPAIR OF LINES-SECOND DEGREE GENERAL EQUATION THEOREM If the equation then i) S ax + hxy + by + gx + fy + c represents a pair of straight lines abc + fgh af bg ch and (ii) h ab, g ac, f bc Proof: Let the
More informationFor more information visit here:
The length or the magnitude of the vector = (a, b, c) is defined by w = a 2 +b 2 +c 2 A vector may be divided by its own length to convert it into a unit vector, i.e.? = u / u. (The vectors have been denoted
More informationGeometry. Class Examples (July 1) Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2014
Geometry lass Examples (July 1) Paul Yiu Department of Mathematics Florida tlantic University c b a Summer 2014 21 Example 11: Three congruent circles in a circle. The three small circles are congruent.
More information1.1 Exercises, Sample Solutions
DM, Chapter, Sample Solutions. Exercises, Sample Solutions 5. Equal vectors have the same magnitude and direction. a) Opposite sides of a parallelogram are parallel and equal in length. AD BC, DC AB b)
More informationMATH 243 Winter 2008 Geometry II: Transformation Geometry Solutions to Problem Set 1 Completion Date: Monday January 21, 2008
MTH 4 Winter 008 Geometry II: Transformation Geometry Solutions to Problem Set 1 ompletion Date: Monday January 1, 008 Department of Mathematical Statistical Sciences University of lberta Question 1. Let
More informationGeometry Problem booklet
Geometry Problem booklet Assoc. Prof. Cornel Pintea E-mail: cpintea math.ubbcluj.ro Contents 1 Week 1: Vector algebra 1 1.1 Brief theoretical background. Free vectors..................... 1 1.1.1 Operations
More informationVectors. 1. Consider the points A (1, 5, 4), B (3, 1, 2) and D (3, k, 2), with (AD) perpendicular to (AB).
Vectors. Consider the points A ( ) B ( ) and D ( k ) with (AD) perpendicular to (AB). (a) Find (i) AB ; (ii) AD giving your answer in terms of k. (b) Show that k = The point C is such that BC = AD. (c)
More informationCO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.
UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose
More informationEngineering Mechanics Statics
Mechanical Systems Engineering- 2016 Engineering Mechanics Statics 2. Force Vectors; Operations on Vectors Dr. Rami Zakaria MECHANICS, UNITS, NUMERICAL CALCULATIONS & GENERAL PROCEDURE FOR ANALYSIS Today
More informationMATH TOURNAMENT 2012 PROBLEMS SOLUTIONS
MATH TOURNAMENT 0 PROBLEMS SOLUTIONS. Consider the eperiment of throwing two 6 sided fair dice, where, the faces are numbered from to 6. What is the probability of the event that the sum of the values
More informationHigher Maths - Expressions and Formulae Revision Questions
Higher Maths - Expressions and Formulae Revision Questions Outcome 1.1 Applying algebraic skills to logarithms and exponentials 1. Simplify fully (a) log 42 + log 48 (b) log 3108 log 34 (c) log 318 - log
More informationQuiz 2 Practice Problems
Quiz Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not essential.
More informationName Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean. A. Definitions: 1.
Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean A. Definitions: 1. Geometric Mean: 2. Right Triangle Altitude Similarity Theorem: If the altitude is
More informationINVERSION IN THE PLANE BERKELEY MATH CIRCLE
INVERSION IN THE PLANE BERKELEY MATH CIRCLE ZVEZDELINA STANKOVA MILLS COLLEGE/UC BERKELEY SEPTEMBER 26TH 2004 Contents 1. Definition of Inversion in the Plane 1 Properties of Inversion 2 Problems 2 2.
More informationPaper 1 Section A. Question Answer 1 A 2 C 3 D 4 A 5 B 6 D 7 C 8 B 9 C 10 B 11 D 12 A 13 B 14 C 15 C 16 A 17 B 18 B 19 C 20 A. Summary A 5 B 6 C 6 D 3
Paper Section A Detailed Marking Instructions : Higher Mathemats 0 Vs Question Answer A C D A B D C B C B D A B C C A B B C 0 A Summar A B C D Page Paper Section B Triangle ABC has vertes A(, 0), B(,)
More informationPaper Reference. 5525/06 Edexcel GCSE Mathematics A 1387 Paper 6 (Calculator) Monday 12 June 2006 Morning Time: 2 hours
Centre No. Paper Reference Surname Initial(s) Candidate No. 5 5 2 5 0 6 Signature Paper Reference(s) 5525/06 Edexcel GCSE Mathematics A 1387 Paper 6 (Calculator) Higher Tier Monday 12 June 2006 Morning
More informationLast week we presented the following expression for the angles between two vectors u and v in R n ( ) θ = cos 1 u v
Orthogonality (I) Last week we presented the following expression for the angles between two vectors u and v in R n ( ) θ = cos 1 u v u v which brings us to the fact that θ = π/2 u v = 0. Definition (Orthogonality).
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 017-18 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500
More informationoo ks. co m w w w.s ur ab For Order : orders@surabooks.com Ph: 960075757 / 84000 http://www.trbtnpsc.com/07/08/th-eam-model-question-papers-download.html Model Question Papers Based on Scheme of Eamination
More information1. Matrices and Determinants
Important Questions 1. Matrices and Determinants Ex.1.1 (2) x 3x y Find the values of x, y, z if 2x + z 3y w = 0 7 3 2a Ex 1.1 (3) 2x 3x y If 2x + z 3y w = 3 2 find x, y, z, w 4 7 Ex 1.1 (13) 3 7 3 2 Find
More informationChapter 2 - Vector Algebra
A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction, and which sums with other vectors according to the Parallelogram Law. A vector can be thought of as an arrow
More information2. In ABC, the measure of angle B is twice the measure of angle A. Angle C measures three times the measure of angle A. If AC = 26, find AB.
2009 FGCU Mathematics Competition. Geometry Individual Test 1. You want to prove that the perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex. Which postulate/theorem
More informationExercises for Unit I (Topics from linear algebra)
Exercises for Unit I (Topics from linear algebra) I.0 : Background Note. There is no corresponding section in the course notes, but as noted at the beginning of Unit I these are a few exercises which involve
More informationAssignment No. 1 RESULTANT OF COPLANAR FORCES
Assignment No. 1 RESULTANT OF COPLANAR FORCES Theory Questions: 1) Define force and body. (Dec. 2004 2 Mks) 2) State and explain the law of transmissibility of forces. (May 2009 4 Mks) Or 3) What is law
More informationBOARD QUESTION PAPER : MARCH 2016 GEOMETRY
BOARD QUESTION PAPER : MARCH 016 GEOMETRY Time : Hours Total Marks : 40 Note: (i) Solve All questions. Draw diagram wherever necessary. (ii) Use of calculator is not allowed. (iii) Diagram is essential
More informationSo, eqn. to the bisector containing (-1, 4) is = x + 27y = 0
Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y
More informationBASIC MATHEMATICS - XII SET - I
BASIC MATHEMATICS - XII Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Candidates are required to give their answers in their own words as far as practicable. The figures in the margin
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH0000 SEMESTER 1 017/018 DR. ANTHONY BROWN 5. Trigonometry 5.1. Parity and Co-Function Identities. In Section 4.6 of Chapter 4 we looked
More informationTopic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths
Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is
More informationFIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, 2013
TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, 2013 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)
More informationApplications of Trigonometry and Vectors. Copyright 2017, 2013, 2009 Pearson Education, Inc.
7 Applications of Trigonometry and Vectors Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 7.4 Geometrically Defined Vectors and Applications Basic Terminology The Equilibrant Incline Applications
More informationEXERCISE - 01 CHECK YOUR GRASP
J-Mathematics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). If ABCDF is a regular hexagon and if AB AC AD A AF AD, then is - (A) 0 (B) (C) (D). If a b is along the angle bisector
More informationStarting with the base and moving counterclockwise, the measured side lengths are 5.5 cm, 2.4 cm, 2.9 cm, 2.5 cm, 1.3 cm, and 2.7 cm.
Chapter 6 Geometric Vectors Chapter 6 Prerequisite Skills Chapter 6 Prerequisite Skills Question 1 Page 302 Starting with the base and moving counterclockwise, the measured side lengths are 5.5 cm, 2.4
More informationCONCURRENT LINES- PROPERTIES RELATED TO A TRIANGLE THEOREM The medians of a triangle are concurrent. Proof: Let A(x 1, y 1 ), B(x, y ), C(x 3, y 3 ) be the vertices of the triangle A(x 1, y 1 ) F E B(x,
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationarxiv: v1 [math.ho] 29 Nov 2017
The Two Incenters of the Arbitrary Convex Quadrilateral Nikolaos Dergiades and Dimitris M. Christodoulou ABSTRACT arxiv:1712.02207v1 [math.ho] 29 Nov 2017 For an arbitrary convex quadrilateral ABCD with
More informationWhich number listed below belongs to the interval 0,7; 0,8? c) 6 7. a) 3 5. b) 7 9. d) 8 9
Problem 1 Which number listed below belongs to the interval 0,7; 0,8? a) 3 5 b) 7 9 c) 6 7 d) 8 9 2 Problem 2 What is the greatest common divisor of the numbers 3 2 3 5 and 2 3 3 5? a) 6 b) 15 c) 30 d)
More informationUnit 2 VCE Specialist Maths. AT 2.1 Vectors Test Date: Friday 29 June 2018 Start Time: Finish Time: Total Time Allowed for Task: 75 min
Unit VCE Specialist Maths AT. Vectors Test Date: Friday 9 June 08 Start Time:.0 Finish Time:.35 Total Time Allowed for Task: 75 min Student Name: Teacher Name: Ms R Vaughan This assessment task will be
More information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More informationConcurrency and Collinearity
Concurrency and Collinearity Victoria Krakovna vkrakovna@gmail.com 1 Elementary Tools Here are some tips for concurrency and collinearity questions: 1. You can often restate a concurrency question as a
More informationπ = d Addition Formulae Revision of some Trigonometry from Unit 1Exact Values H U1 O3 Trigonometry 26th November 2013 H U2 O3 Trig Formulae
26th November 2013 H U1 O3 Trigonometry Addition Formulae Revision of some Trigonometry from Unit 1Exact Values 2 30 o 30 o 3 2 r π = d 180 1 1 45 o 45 o 2 45 o 45 o 1 1 60 o 60 o 1 2 1 180 x x 180 + x
More information1 / 23
CBSE-XII-017 EXAMINATION CBSE-X-008 EXAMINATION MATHEMATICS Series: RLH/ Paper & Solution Code: 30//1 Time: 3 Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question
More information2 and v! = 3 i! + 5 j! are given.
1. ABCD is a rectangle and O is the midpoint of [AB]. D C 2. The vectors i!, j! are unit vectors along the x-axis and y-axis respectively. The vectors u! = i! + j! 2 and v! = 3 i! + 5 j! are given. (a)
More informationSection 13.4 The Cross Product
Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions
More informationProblems and Solutions: INMO-2012
Problems and Solutions: INMO-2012 1. Let ABCD be a quadrilateral inscribed in a circle. Suppose AB = 2+ 2 and AB subtends 135 at the centre of the circle. Find the maximum possible area of ABCD. Solution:
More informationThe Cross Product of Two Vectors
The Cross roduct of Two Vectors In proving some statements involving surface integrals, there will be a need to approximate areas of segments of the surface by areas of parallelograms. Therefore it is
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More informationDepartment of Physics, Korea University
Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an
More informationMULTIPLE PRODUCTS OBJECTIVES. If a i j,b j k,c i k, = + = + = + then a. ( b c) ) 8 ) 6 3) 4 5). If a = 3i j+ k and b 3i j k = = +, then a. ( a b) = ) 0 ) 3) 3 4) not defined { } 3. The scalar a. ( b c)
More informationVECTOR ALGEBRA. 3. write a linear vector in the direction of the sum of the vector a = 2i + 2j 5k and
1 mark questions VECTOR ALGEBRA 1. Find a vector in the direction of vector 2i 3j + 6k which has magnitude 21 units Ans. 6i-9j+18k 2. Find a vector a of magnitude 5 2, making an angle of π with X- axis,
More information