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 vector of a point P(x, y, z w.r.t. origin 0(0,0,0 is denoted y OP Ç, where OP Ç = xiˆ È yj ˆ È zkˆ É and OP = x Ê y Ê z l The angles Ë, Ì, Í made r Î with positive direction of X, Y and Z-axies respectivily are called direction angles and cosines of these angles are called direction cosines of r Î usually denoted as l = cos Ë, m = così and n = cos Í where l + m + n = 1. l The numers a,, c, propotional to l, m and n are called direction ratios i.e. a Ï Ï c. l m n l If two vector a Æ and Ð are represented in magnitude and direction y the sides of a triangle taken Ò in order, then their sum a Ñ Ò is represented in magnitude and direction y the third side of triangle taken in opposite order. This is called triangle law of addition of vectors. l If a Æ is any vector and Ó is a scalar, then Ó a Æ is a vector, collinear with a Æ and Ôa Õ = Ô a Õ. l Any vector a Æ can e written is a Æ Ö = a aˆ where â is a unit vector in the direction of â. l If A(x 1, y 1, z 1 and B(x, y, z e any two points in space, then AB = (x x 1 î + (y y 1 ˆ j ( z z kˆ. l 1 If a Æ and Ð e the position vectors of points A and B, and C is any point dividing AB internally in m : n, then position vector c Æ of C is given as c Æ = m Ø na. If C divides AB in m : n externally, m Ø n then C m na =. m n l Scalar Product (Dot Product of two vectors a Æ and Ð Ù is denoted y a. Ù Ü Ü Ü Ü and is defined as a. Ú a cosû, where Ý is the angle etween a Æ and Ð (0 ÞÝ Þ l Dot product of two vectors is commutative i.e., à à à à a. ß. a l For unlike parallel vectors a Æ and Ð Ù, a. Ù = a á á â â, so a. a = a ã l For unlike parallel vectors a Æ and Ð Ù, a. Ù = a ä ä 9 1
è è è è è è l If a. å 0 æ a å 0 or å 0 or a ç. l iˆ. ˆj é ˆj. kˆ é kˆ. iˆ é 0, iˆ. iˆ é ˆj. ˆj é kˆ. kˆ é 1 ì l If ˆ ˆ ˆ ì a ê a1i ë a j ë a3k and ˆ ˆ ˆ ï ï ê 1i ë j ë 3k, then a. í a1a î 1 î c1c. l Projection of a vector a ñ on ò ó ó a. = ó l Projection of a vector a ô along õ ö ü ü a. = ü ˆ. øú ùû l Cross product (Vector product of two vectors a ô and õ ý is denoted as a ý and is defined as a þ þ ÿ = a ÿ sinð ˆ, where ð is the angle etween a ñ and ò (0 ð and ˆ is a unit vector perpendicular to oth a ñ and ò such that a ñ, ò and ˆ form a right handed system. l Vector product of two vectors a ñ and ò is not commutative i.e., a a, ut a a. l If a O a O or O, or a. l a. a O, So iˆ iˆ O ˆj ˆj kˆ kˆ. l If ˆ ˆ ˆ a ê a1i ë a j ë a3k and ê ˆ ˆ ˆ 1i ë j ë 3k, then a iˆ ˆj kˆ = a1 a a3 1 3 l If ð is angle etween two non zero vector a and, then sinð = a a l a is the area of that parallelogram where adjacent sides are vectors a and. 1 l a is the area of that parallelorgam whose diagonals are a and. l If a, and c forms a triangle, then area of the triangle = 1 a = 1 c = 1. c a l If a, and c are position vectors of vertices of a triangle, then area of triangle = 1 a c c a Scalar triple product of vectors The scalar triple product of three vectors a c 9
Geometrical Interpretation : ( a. c represents the volume of parallelepiped whose coterminous edges are represented y a and c. Scalar triple product of three vectors remains unchanged as long as their cyclic order remains unchanged. ( a. c = ( c. a = ( c a. or [ a c] = [ c a] = [ c a ] Scalar triple product changes its sign ut not magnitude, when the cyclic order of vectors is changed. [ a c] = [ a c ] Scalar triple product vanishes if any two of its vector are equal. Scalar triple product vanishes if any two if its vector are parallel or collinear. Necessary and sufficient condition for three non zero non collinear vectors a and c to e coplanar is that [ a c] = 0 Scalar triple Product in terms of component a a i ˆ a ˆ j a k i ˆ ˆ j k Let = 1 3 ˆ 1 3 ˆ c c i c j c k ˆ ˆ ˆ 1 3 then [ a c] = a a a 1 3 1 3 c c c 1 3 Question for Practice Very Short Answer Type Questions (1 Mark 1. Write direction cosines of vector iˆ ˆj kˆ. For what value of x the vectors iˆ 3 ˆj 4kˆ and xiˆ 6 ˆj 8kˆ are collinear 3. Write the projection of the vector iˆ ˆj on the vector iˆ ˆj 4. Write the angle etween two vectors a and with magnitudes 3 and respectively having a. 6 5. If a. a 0 and a. 0 = 0 then what can e concluded aout vector 6. What is the cosine of the angle which the vector iˆ ˆj kˆ makes with z axis 7. Write a vector of Magnitude 3 units in the direction of vector iˆ ˆ j kˆ 9 3
8. If a = iˆ ˆj 3kˆ and = 6iˆ ˆj 9kˆ and a 9 4 find the value of 9. For what value of a = iˆ ˆj kˆ and = iˆ ˆj kˆ are prependicular to each other 10. If a is a unit vector such that a i ˆj find a. i 11. a = = 7, a! 3iˆ " ˆj " 6kˆ write the angle etween a and 1. If a, represent Diagonal of rhomus, then write the value of a. 13. Write unit vector in the direction of a # where a $ iˆ % ˆ j kˆ, & iˆ ' ˆj ' kˆ 14. Write a, If two vector a and are such that a =, = 3, a. = 4 15. If a and are unit vectors such that a is also a unit vector. Find the angle etween a and 16. What is the value of ( x a.( x ( a 15 x If for unit vector a 17. Write the value of iˆ.( ˆj kˆ * ˆj.(( kˆ iˆ * kˆ.( ˆj i 18. Find the value of so that the vectors a = iˆ 3 ˆj kˆ and = iˆ ˆj 3kˆ and c + ˆj, -kˆ are coplanar 19. Find the value [ iˆ ˆj kˆ ] 0. Find the volume of parallelepiped whose coterminous edges are represented y the vectors a = iˆ ˆj 3kˆ and = 3iˆ 7 ˆj 4kˆ and c. iˆ 5 ˆj 3kˆ Short Answer Type Questions (4 Marks 1. If a,, c are unit vectors such that a. / a. c / 0 and the angle etween and c is 0 6 then prove that a = 1 ( c. Find a vector of magnitude 5 units, prependicular to each of the vectors ( a * and ( a where a iˆ 3 ˆj 3 kˆ, 4 iˆ 5 ˆj 5 3kˆ 3. If a,, c are vectors such that a,. a. c a 6 a c and a 0 then prove that 7 c 4. If a iˆ 3 ˆj 3 kˆ and 8 ˆ j kˆ find a vector c such that a c 9 and a. c : 3 5. Using vectors find the area of triangle with vertices A(1, 1, B(, 3, 5 and (1, 5, 5 6. If a,, c are the position vectors of the vertices A, B, C of ABC respectively, find an expression for the area of ABC and Hence deduce the condition for the points A, B, C to e collinear 7. Show that the area of gm having Diagonals 3 iˆ * ˆj kˆ and iˆ 3 ˆj * 4kˆ is 5 3 sq unit.
8. Let a,, c e three vectors of magnitude 3, 4, 5 units respectively. If each of these is ; to the sum of other two vectors, then find a < < c 9. If a, and c are three vectors such that a = = c > 0 a = 3 = 5 c = 7 find the angle etween a and 10. If a and are unit vectors and? is the angle etween them, then show that @ 1 (i sin A ˆ a B 1 (ii cos C a ˆ D 11. If a unit vector a makes 4 and 3 with x axis and y axis respectively and an otuse angle? with z axis, then find? and the components of a along axis. 1. If a and are unit vectors such that a 4 and 10a E 8 are perpendicular to each other. Find the angle etween a and 13. Find a vector a such that.( ˆ ˆ a i F j G.( ˆ ˆ a i j H 3 a. kˆ I 0 14. If a makes equal angles with the coordinate axes and has magnitude 3, then find the angle etween a and each of three coordinate axes. 15. A girl walks 4 km west wards, then she walks 3 km is a direction 30 east of north and stops. Determine the girls displacement from her initial point of departure 16. Prove the following (i [ a J, J c, c J a] K [ a c] (ii [ a, c, c a] H 0 17. Show that the points with position vectors 6 iˆ 7 ˆj,16 iˆ 19 ˆj 4 kˆ, 3 iˆ 6k ˆ and iˆ 5 ˆj L 10kˆ are coplanar. 18. If a M c d, a c M d show that a d is to c 19. Let ˆ ˆ 3 ˆ ˆ a N i j N i k c N 7 iˆ kˆ find a vector d which is ; to oth a and and c. d O 1 0. If a P P a Q P 1 then find a 1. Prove a =. a a a. a... If a P P 5 a P 8 find a. 3. Let u, v u + v R w S o If u T 3 v T 4 w T 5 find u. v U v. w U wu. 9 5
V V V V X X X X X X X 4. In a Regular hexagon ABCDEF if AB W a BC W then express CD,DE,EF,FA,AC,AD,AE, and CE Y is terms of a and ] ] 5. Given a Z 3 iˆ ˆj and [ iˆ \ ˆj 3kˆ express ^ 1 _ where, is parallel to a and ] is ` a Answers 1. 1 1 1,, 3 3 3 3. 0 4.. x = 4 5. a a 6. cos = 1 7. 3( iˆ c ˆj kˆ 8. 3 9. d = 1 10. 0 11. 6 1. 0 1 13. (iˆ e ˆj e kˆ 3 14. 5 15. 16. 4 17. 1 18. d = 1 19. 1 0. 39 cuic unit Answer of four marks question 4 1. Given a. f 0 and a. c g 0 h is ` to oth and c Also a is a unit vector i = c c 1 1 But c j c sin j 11 j 6 5. ( iˆ k ˆj kˆ 6 3. a. l a. c m a. n a. c l 0 m a.( c l o a o o or c p o or a q ( c (1 ut a o (given a = a c h a a c r o h ( a c s o a o o or c p o or a c ( ut a o from (1 and ( we get t c ( a u c and a c can not held simultaneously 9 6
r e a o f ABC 4. 5 ˆ ˆ c v i w j w k ˆ 3 3 3 5. 1 61 6. A = 1 x x C( c AB AC = 1 ( ( a c a = 1 c a a c y a a = 1 a z c z c a A( a ( a a { o a a Point A, B, C to e collinear if area of ABC = 0 a } c } c a ~ 0 1 7. Hint Area of 11gm 1 d d d 1 & d are the vectors along diagonal 8. 5 9. a c 0 a c ( a.( a ƒ ( c.( c a a. c a a cos c B( 3 + 5 + 3 5 cos = 7 = 60 11. = ˆ 1, 1 1,, 3 i j k 1. angle etween a and is 10. 5 1 13. a Š iˆ j 14. Let a x ˆ ˆ ˆ 1i Œ y1 j Œ z1k Similarly a = 3 3 3 cos 3 1 1 1 cos = x y z = 3 a.ˆ i x = 1 a iˆ 3 x 1 = 3cos y 1 = 3cos z 1 = 3cos 1 cos cos 3 9cos Ž 9cos Ž 9cos Ž = 3 1 1 3 9 7
15. OB = OA AB OA = 4iˆ In AMB AM = AB cos60 MB = ABsin60 AM = 3 AB š š = AM MB 3 3 3 AB = iˆ œ ˆj Girls displacement from initial point of departure 3 3 3 ž5 3 3 ž4 i Ÿ i Ÿ j i Ÿ j MB = 3 3 19. 1 1 3 i ˆ ˆ j k ˆ 4 4 4. 6 3. 5 4. CD a DE a EF FA ª a 5. š 1 = 3 iˆ 1 ˆj = 1 iˆ 3 ˆ j 3kˆ AC a «AD AE a CE a 9 8