After studying this unit, you should be able to: define scalar product or dot product of vectors; find angle between two vectors;

Transcription

1 Vectors and Three Dimensional Geometry UNIT 2 VECTORS 2 Structure 2.0 Introduction 2.1 Objectives 2.2 Scalar Product of Vectors 2.3 Vector Product (or Cross Product) of two Vectors 2.4 Triple Product of Vectors 2.5 Answers to Check Your Progress 2.6 Summary 2.0 INTRODUCTION In the previous unit, we discussed vectors and scalars. We learnt how to add and subtract two vectors, and how to multiply a vector by a scalar. In this unit, we shall discuss multiplication of vectors. There are two ways of defining product of vectors. We can multiply two vectors to get a scalar or a vector. The former is called scalar product or dot product of vectors and the latter is called vector product or cross product of vectors. We shall learn many applications of dot product and cross product of vectors. We shall use dot product to find angle between two vectors. Two vectors are perpendicular if their dot product is zero. Dot product helps in finding projection of a vector onto another vector. The cross product of two vectors is a vector perpendicular to both the vectors. If cross product of two vectors is zero then the two vectors are parallel (or collinear). Cross product of vectors is also used in finding area of a triangle or a parallelogram. Using the two kinds of products, we can also find product of three vectors. Many of these products will not be defined. In this unit, we shall discuss the two valid triple products, namely, the scalar triple product and the vector triple product. 2.1 OBJECTIVES After studying this unit, you should be able to: define scalar product or dot product of vectors; find angle between two vectors; 28 find projection of a vector on another vector;

2 define cross product or vector product of vectors; Vectors - II use cross product to find area of a parallelogram vector product of vectors; define scalar triple product and vector triple product of vectors. 2.2 SCALAR PRODUCT OF TWO VECTORS Definition : The scalar product or the dot product of two vectors and, denoted by. is defined by. where is the angle between the vectors and. Also the scalar product of any vector with the zero vector is, by definition, the scalar zero. It is clear from the definition that the dot product. is a scalar quantity. Sign of the Scalar Product If and are two non zero vectors, then the scalar product. cos is positive, negative or zero, according as the angle acute, obtuse or right. In fact,, between the vectors is is acute cos 0 ab. 0 is right cos 0 ab. 0 is obtuse cos 0 ab. 0 Also, note that if and and are non zero vectors then. 0 if and only if and are perpendicular (or orthogonal) to each other. If. is any vector, then the dot product.., of with itself, is given by. cos 0 2 Thus, the length of any vector is the non negative square root, i.e., of the scalar product, i.e.,, Angle between two Vectors If is an angle between two non zero vectors and, then. cos 29

3 Vectors and Three Dimensional Geometry So, the angle between two vectors is given by Properties of Scalar Product 1. Scalar product is cumulative, i.e.,.. for every pair of vectors and. 2.. ( ) (. ) and ( ). ( ). for every pair of vectors and. 3. ( λ 1 ). (λ 2 ) ( λ 1 λ 2 ) (. ) where and are vectors and λ 1, λ 2 are scalars 4. (Distributivity).( + ). +. The following identities can be easily proved using above properties. (i) ( + ). ( ) a 2 b 2 (here, a 2. 2 ) (ii) ( + ) 2 a b 2 (iii) ( ) 2 a b 2 If, and are mutually perpendicular unit vectors, then... 1 and (1) If and be two vectors in component form, then their scalar product is given by a 1i ˆ. ( biˆ 1 + b ˆ ˆ ˆ ˆ 2 j b ˆ ˆ ˆ 3k) a2i ( bi 1 + b2 j b3k ) a3k( bi 1 + b2 j b3k ) (using distributivity) a b ( j. kˆ ) a b ( kˆ. iˆ ) a b ( kˆ. j ) a b ( kˆ. kˆ ) (using properties) a 1 b 1 + a 2 b 2 + a 3 b 3 (using (1)) Thus. a 1 b 1 + a 2 b 2 + a 3 b 3. Projection of a Vectors on another Vector Let be two vectors A 30 O L B Figure 1

4 Drop a perpendicular form A on OB as shown in Figure 1. The projection of Vectors - II on is the vector. If is the angle between and, then projection of and has length cos and direction along unit vector. Thus, projection of on ( cos ) a. b a. b b b 2 b b a. b The scalar component of project of a on b is. Similarly, the scalar b a. b component of projection of b on a a Example 1: Find the angle between the following pair of vectors. (a) + (b) Solution: (a). (2 + 3 ). ( + 4 ) ( 1) (2 + 3 ). (2 + 3 ) a. ( +4 ). ( +4 ) ( 3) ( 3) 26 a (b) Here,. ( ). ( ) ( 2)

5 Vectors and Three Dimensional Geometry 3. Example 2 : Show that is perpendicular to, for any two non zero vectors. Solution : We know that two vectors are perpendicular if their scalar product is zero. ( ). ( ) ( ) +. ( ) (using distributivity) ( ) ( ) + ( ) ( ) ( ) + ( ) 0 so, the given vectors are perpendicular. Example 3 : Find the scalar component of projection of the vector a 2iˆ 3 ˆj 5 kˆ on the vector b 2 i 2 ˆj kˆ. Here, and b 2 ( 2) ( 1) 3 Example 4 : Show that the diagonals of a rhombus are at right angles. Solution : Let A B C D be a rhombus (Figure 2 ) D C 32 A (Figure 2) B

6 ABCD being a rhombus, we have Vectors - II ( + ). ( ) ( AB AD ) Example 5 : For any vectors and, prove the triangle inequality + + Solution : If or, then the inequality holds trivially. So let 0. Then, ( + ) 2 ( + ). ( + ) ( ) 2 2 cos 2 where is the angle between and ( cos 1 ) ( ) 2 Hence Remark : Let and, then + + C A B Figure 3 As shown in figure 3 inequality says that the sum of two sides of triangle is greater than the third side. If the equality holds in triangle inequality, i.e., + + Then AC AB + BC showing that the points A, B and C are collinear. Check Your Progress 1 1. If and + 5 then show that + and are perpendicular. 33

7 Vectors and Three 2. Find the angle between the vectors Dimensional Geometry (a) (b) Find the vector projection of on where and Also find the scalar component of projection of vector on. 4. Prove the Cauchy Schwarz Inequality. 5. Prove that ab 2.3 VECTOR PRODUCT (OR CROSS PRODUCT) OF TWO VECTORS Definition : If and are two non zero and non parallel (or equivalently non collinear) vectors, then their vector product is defined as sin where is the angle between and (0 < < ) and is a unit vector perpendicular to both and such that, and form a right handed system. If and are parallel (or collinear) i.e., when 0 or, then we define the vector product of and to be the zero vector i.e.,. Also note that if either or,then Properties of the Vector Product 1. since 0 2. Vector product is not commutative i.e.,. However, We have sin where, and form a right handed system and a sin where, and ight handed system. So the direction of is opposite to that of Hence, sin sin. 3. Let from a right handed system of mutually perpendicular unit vectors in three dimensional space. Then and Also, Figure 4 34

8 4. Two non zero vectors and are parallel if and only if 5. Vector product is distributors over addition i.e., if, and are three vectors, then (i) ( ). (ii) ( ). 6. If is a scalar and and are vectors, then ( ( Vectors - II Vector Product in the component form Let a 1 a 2 a 3 and b 1 b 2 b 3 Then (a 1 a 2 a 3 ) (b 1 b 2 b 3 a 1 b 1 ( ) a 1 b 2 ( ) a 1 b 3 ( ) a 2 b 1 ( ) a 2 b 3 ( ) a 2 b 3 ( ) a 3 b 1 ) a 3 b 2 ( ) a 3 b 3 ( ) a 1 b 2 a 1 b 3 ( a 2 b 1 ( a 2 b 3 a 3 b 1 a 3 b 2 ( (a 2 b 3 a 3 b 2 ) (a 1 b 3 a 3 b 1 ) (a 1 b 2 a 2 b 1 ) Example 6: Find if and 3 Solution : We have (3 2) (3 1) + (2 1) 2 Example 7: Find a unit vector perpendicular to both the vectors a 4iˆ ˆj 3 kˆ and b 2 i ˆj 2 kˆ Solution : A vector perpendicular to both and is (2 3) ( 8 6) (4 2) 2 3. So the desired unit vector is 35

9 Vectors and Three Dimensional Geometry Example 8 : Prove the distributive law a ( b c) a b a c using component form of vectors. Solution : Let + + Now, + + and + +. So, ( + ( + ) +. Example 9: Show that 2 ( Solution : 2 sin 2 ( sin ) (1 cos 2 ) a 2 b 2 ( a 2 2 cos 2 ) 2 2 ( aa) ( bb) ( a. b) Example 10 : Find if Solution : Here 12 10, 2,

10 Area of Triangle Vectors - II Let, be two vectors and let be the angle between them ( O < < ). Let O be the origin for the vectors as shown in figures (Fig. 5) below and let, B D θ Draw BC OA. O C A Figure 5 Then BC OB Sin b sin Thus, if and represent the adjacent sides of a triangle, then its area is given as Area of a Parallelogram In above figure, if D is the fourth vertex of the parallelogram formed by O, B, A then its area is twice the area of the triangle OBA. Hence, area of a parallelogram with and as adjacent sides Example 11: Find the area of C( 1, 2, 3). ABC with vertices A (1,3,2), B (2, 1,1) and Solution : We have (2 1) ( 1 3) (1 2) 4 and ( 1 1) (2 3) (3 2) 2 37

11 Vectors and Three Dimensional Geometry Example 12 : Show that ( ) ( ) 2( ). Interpret the result geometrically. Solution : ( ) ( ) ( ) ( ) ( ) Let ABCD be a parallelogram with and. Then area of parallelogram D C b A a B Figure 6 Also, diagonal + and diagonal ( ) ( ) area of parallelogram formed by and. Thus, the above result shows that the area of a parallelogram formed by diagonals of a parallelogram is twice the area of the parallelogram. Check Your Progress 2 1. Find a unit vector perpendicular to each of the vector ( + ) and ( ), where 2 4 and 2 2. Show that ( ) ( ) ( ) 38

12 3. For Vectors 2 Compute ( and ( ). is ( ( ). 4. If the vectors, and satisfy + + 0, then prove that 5. Find the area of a parallelogram whose diagonals are Vectors - II 2.4 TRIPLE PRODUCT OF VECTORS Product of three vectors may or may not have a meaning. For example, (. as is a scalar and dot product is defined only for vectors. Similarly, (. has no meaning. The products of the type ( and are meaningful and called triple products. The former is a vector while the latter is a scalar. Scalar Triple Product Definition : Let, and be any three vectors. The scalar product of. ) is called scalar triple product of, and and is denoted by [, ]. Thus [, ].(. It is clear from the definition that [, ]. Is a scalar quantity. Geometrically, the scalar triple product gives the volume of a parallelepiped formed by vectors, as adjacent sides. Scalar Triple Product as a determinant Let and Then.( ( 39

13 Vectors and Three Dimensional Geometry Thus,.( Note : We can omit the brackets in.( (. is meaningless. because Properties of Scalar Triple Product 1. [, [ [ ] [, This is clear if we note the properties of a determinant as [, can be expressed as a determinant. 2. In scalar triple product.(, the dot and cross can be interchanged. Indeed,.( [ [ 3. [ pa, qb, rc] pqr[ a, b, c] where p, q and r are scalars. Again it is clear using properties of determinants. 4. If any two of are the same then [ For example, [, Coplanarity of three vectors Theorem : Three vectors are coplanar if and only if [ Proof : First suppose that are coplanar. If are parallel vectors, then If are not parallel, then being perpendicular to plane containing is also perpendicular to because Conversely, suppose that [, 0. If are both non zero, then are perpendicular as their dot product is zero. But is perpendicular to both and hence must lie in a plane, i.e., they are coplanar. If then are coplanar as zero vector is coplanar with any two vectors. If, must be coplanar., then Note : Four points A, B, C and D are coplanar if the vectors, and are coplanar. 40

14 Example 13 : If , and find [,, Vectors - II Solution : [,,. ( 7(0 16) + 2(0 4) + 3(8 + 4) Example 14 : Find the value of λ for which the vectors 4 + λ 2 +, and are coplanar, Solution : If, and are coplanar, we have. ( 0 i.e., 1( 6 3) + 4(3 2) + (3 + 4) 0 Example 15 : If,, are coplanar then prove that, and are also coplanar. Solution : Since,, are coplanar, [,, Now [, ] ( ).( ) ( + ).[( + ( ] ( + ).[ ] ( as 0.( a.( ) +.( ) + ( + ( ) + [,, [,, + 2 [,, (using property (4) 0 41

15 Vectors and Three Vector Triple Product Dimensional Geometry Definition : Let,, and any three vectors. Then, the vectors ( ) and ( ) are called vector triple products. It is clear that, in general ( ) ( ) Note that ( ) is a vector perpendicular to both and. And also is perpendicular to both and. Thus ( ) lies in a plane containing the vectors and, i.e., ( ), and are coplanar vectors. We now show that for any three vectors, ( ) ( ( Let and Then ( + ( + ( ( [ ( + ( + [ ( Also, (1) ( ( [ ( [ ( [ ( [ ( [ ( [ ( (2) 42

16 From (1) and (2), we have ( ( ( Vectors - II It is also clear from this expression that ( is a vector in the plane of and Now ( [ ( ) ( ) ( ) So, ( is a vector in the plane of. Theorem : Let if and only if be any three vectors. Then are collinear. Proof : First suppose Now, ( and ( ( So, ( a. b) c ( b. c) a Conversely, suppose that. Then ( and ( So, ( ( ( ( i.e., ( Example 16 : Show that Solution : ( (. ) ( ) + ) ( since dot product is commutative. Example 17 : Let be any four vectors. Then prove that (i) ( ) [ ] c [ (ii) ( ) [ ] [ 43

17 Vectors and Three Solution : (i) Let Then Dimensional Geometry ( ) ) ( ( [ [ [ (ii) Let. Then ( ) ( ( [( r ( ( ( ( ) ( ) Example 18 : Prove that [, ] [ ] [,, a Solution : [,, a ). [ ) ( )[ ) ) ). [ ) a c ] ( ) ) [ ) ] [ a )][ )] [ ] 2. Example 19 : For vectors Solution : We have and verify that ) )

18 So, ( Vectors - II Also, (1) ( 2 +.( ( ) 4 ( and ( +2 +.( ( ) + 2 Thus, ( ) ( ) 6 (2) Hence, from (1) and (2) Check Your Progress 3 ) ( ) ( ) 1. For vectors +, find [, ]. 2. Find the volume of the parallelepiped whose edges are represented by 3, Show that the four points having position vectors +, i are coplanar. 4. Prove that ( ). ( ) (. )(. ) ( )(. ) 5. For any vector prove that 6. Prove that ( ) + ( ) + ( ) 2 [ ( )] (. ) ( ) 2.5 ANSWERS TO CHECK YOUR PROGRESS Check Your Progress 1 1. Here, and So, ( + ). ( ) ( Hence + and + are perpendicular vectors. 45

19 Vectors and Three 2. (a) Here,. 3 ( 4) + ( 1) Dimensional Geometry (b) Here,. 1( 2) + ( 1) 2 + ( 2) 4 12 θ 3. Vector Projection of on ab. 2 b b 4. The inequality holds trivially if ( a b) c a ( b c) 0, b 0 Now, cos Hence b ( dot product is commulative)

20 Check Your Progress 2 Vectors - II 1. Here Let. Then is vector perpendicular to both Now ( 6 + 6) 12 A unit vector in the direction of is So, is a unit vector perpendicular to both 2. ) a ( ) (using distributivity) ( and 0 3. Also, 5 Clearly, ( 47

21 Vectors and Three Dimensional Geometry 4. i.e., + i.e., + 0 ( i.e., i.e.,..(1) Similarly, i.e., + i.e., i.e.,. (2) From (1) and (2), we have 5. Let ABCD be the parallelogram D C A B Figure 7 So area of parallelogram ABCD 48

22 Now, Vectors - II AB AD 5 (2 3) (1+ 6) + ( 1 4) Hence, the area of parallelogram ABCD 5 Check Your Progress ( (2 ). (3 ) Volume of Parallelopiped ( Now, ( 7 Volume Let So, +12 OC

23 Vectors and Three Dimensional Geometry AB, BC and CD are coplanar. Hence A, B, C and D are coplanar. 4. Let Then ( ). ( ) r.( ) ( r ). (( ) ). [( ( ) (. ) ( ) ( ) 5. L.H.S. ( ) ( ) ( ) [ ( ) i [ ( ) ) i ) Let So, Similarly, L.H.S ( ( ( 0 ( ( 50

24 2.6 SUMMARY Vectors - II This unit discusses various operations on vectors. The binary operation of scalar product is discussed in section 2.2. In the next section, the binary operation of vector product (also, called cross product) is illustrated. Finally in section 2.4, the ternary operation of tripe product of vectors is explained. Answers/Solutions to questions/problems/exercises given in various sections of the unit are available in section