Tiêu đề XII - maths - chapter 9 - DOT PRODUCT OF VECTORS (11.03.2015)(35-61).pdf ✅

3 6 DOT PRODUCT OF VECTORS JEE MAINS - C.W - VOL - I JEE MAINS - VOL - I  Let a and b be two nonzero vectors. Then Component vector of b on a (or) orthogonal projection of b on a is 2 (b.a)a a WE-2: Orthogonal projection of b = 2 3 6 2 2 i j k ona i j k      is Sol: Orthogonal projection 2 [(2 3 6 ). ( 2 2 )]( 2 2 ) 2 2 i j k i j k i j k i j k          (2 6 12) 8 ( 2 2 ) ( 2 2 ) 9 9 i j k i j k          Component vector of a on b (or) orthogonal projection of a on b is 2 (a.b)b b WE-3: If a i j k b i j k       2 2 , 5 3 then orthogonal projection of a on b is Sol: Orthogonal projection of a on b is 2 (a.b)b b  (10 3 2)(5 3 ) 9(5 3 ) 25 9 1 35       i j k i j k      The orthogonal projection of b in the direction perpendicular to that of a is 2 (b.a)a b a  WE-4: The orthogonal projection of b i j k    3 2 5 on a vector perpendicular to a i j k    2 2 is Sol: Orthogonal projection of b on a     3 2 5 i j k (3 2 5 ).(2 2 ) 4 1 4 i j k i j k       (2 2 ) i j k   6 2 10 3 2 5 9 i j k       (2 2 ) i j k      (3 2 5 ) i j k 2 (2 2 ) 3    i j k 13 4 11 3 i j k     The length of the orthogonal projection of b on a is (a.b) a  The length of the orthogonal projection of a on b is (a.b) b WE-5: The length of orthogonal projection of a i j k    2 3 on b i j k    4 4 7 is Sol: The length of the orthogonal projection of a on b is (a.b) 27 8 12 7 3 b 16 16 49 9         The scalar product is commutative i.e., a . b = b . a  The scalar product is distributive over vector addition i.e., a .( b + c ) = a . b + a . c , ( b + c ). a = b . a + c . a  l a b a l b l a b . . .       where l is a scalar  a a a b a b a b a b . 0; . ;      a b a b     Cauchy schwartz in equality : Let 1 2 3 1 2 3 a a a and b b b , , , , be real numbers. Then   2 a b a b a b 1 1 2 2 3 3       2 2 2 2 2 2 a a a b b b 1 2 3 1 2 3     and equality holds If 1 2 3 1 2 3 a a a b b b    2 2 2 a b a b 2a.b      2 2 2 a b a b 2a.b      2 a b c    2 2 2 a b c 2(a.b b.c c.a)     
3 7 JEE MAINS - VOL - I DOT PRODUCT OF VECTORS  Let 1 1 1 l ,m ,n be the direction cosines of a and let 2 2 2 l ,m ,n be the direction cosines of b and let (a, b)  then Cos l l m m n n     1 2 1 2 1 2  The vector equation to the plane which is at a distance of p units from the origin and nˆ is a unit vector perpendicular to the plane is r n p . ˆ   If the origin lies on the plane then its equation is r n. 0   The vector equation of a plane passing through the point A a  and perpendicular to the vector n is r a n   . 0 W.E-6 : The vector equation of the plane passing through the point 3, 2,1   and perpendicular to the vector 4,7, 4  is Sol. r i j k i j k       3 2 . 4 7 4 0        r i j k . 4 7 4   3 2 . 4 7 4 i j k i j k           12 14 4      r i j k . 4 7 4 6    In a parallelogram, if its diagonals are equal then it is a rectangle.  In a parallelogram, the sum of the squares of the diagonals is equal to the sum of the squares of the sides.  If F be the force and s be the displacement inclined at an angle  with the direction of the force, then work done F S.  If a constant force F acting on a particle displaces it from A to B, then work done, W F AB  .  If F is the resultant of the forces 1 2 , ...... F F Fn then work done in displacing the particle from A to B is W F F F AB      1 2 .... . n   If a number of forces are acting on a particle, the sum of the work done by the seperate forces is equal to the work done by the resultant force.  A line makes angles , , , , with the four diagonals of a cube then 2 2 cos cos     2 2 cos cos 4 / 3      If r is any vector then r r i i r j j r k k    ( . ) ( . ) ( . ) .  If a b, are two vectors then i) a a. 0  ii) a b a b . | || |  iii) a b a b    iv) a b a b     The cartesian equation of the plane passing through the point 1 1 1 A x y z ( , , ) and perpendicular to the vector m ai bj ck    is 1 1 1 a x x b y y c z z ( ) ( ) ( ) 0       .  The equation of the plane passing through the point 1 1 1 A x y z ( , , ) and whose normal has d.r.s a,b,c is 1 1 1 a x x b y y c z z ( ) ( ) ( ) 0       .  Angle between any two diagonals of a cube is 1 cos (1/ 3)  .  Angle between a diagonal of a cube and a diagonal of a face of the cube which are passes through the same corner is 1 cos 2 / 3  .  Angle between a diagonal of a cube and edge of a cube is 1 1 cos 3        Angle between a line and a plane :  i) The angle between a line and a plane is the complement of the angle between the line and normal to the plane. If  is the anlge between a line r a tb   and a plane r m d .  then   0 . cos 90 sin b m b m      . WE-7: The angle between the line r i j k t i j k        ( 3 3 ) (2 3 6 ) and the plane r i j k .( ) 5     is
3 8 DOT PRODUCT OF VECTORS JEE MAINS - C.W - VOL - I JEE MAINS - VOL - I Sol: (2 3 6 ).( ) sin 2 3 6 i j k i j k i j k i j k              2 3 6 1 1 1 sin 4 9 36 1 1 1 3 3                    If  is the angle between the line 1 1 1 x x y y z z l m n      and the plane ax by cz d     0 then i) 2 2 2 2 2 2 sin al bm cn a b c l m n         ii) If the line is perpendicular to the plane then l m n a b c   . iii) If the line is parallel to the plane then al bm cn    0  The perpendicular distance of the plane r n a n . .  . from the origin is a n. n  Angle between the planes 1 1 2 2 r n p r n p then . , . ,   1 2 1 2 n n. Cos n n   WE-8: Angle between the planes r i j k .(2 ) 3    and r i j k .( 2 ) 4    is Sol: Let a i j k    2 and b i j k    2 If  is the angle between the planes then . cos cos( , ) a b a b a b     2 1 2 3 1 60 4 1 1 1 1 4 6 2             1. If a b a b    then the angle between the vectors a,b is 1) Acute Angle 2) Obtuse Angle 3) Right Angle 4) 450 2. R r  is any point on the semi-circle P p  and Q q  are the position vectors of diameter of that semicircle. Then PR QR . is equal to 1) 1 2) 0 3) 3 4) Not defined 3. The non-zero vectors a b c , , are related by a b  8 and c b  7 , then the angle between a c & is (AIE-2008) 1) 0 2) 4  3) 2  4)  4. If 0 0 ( , ) 0 180 , a b or then a b are  1) Perpendicular 2) Parallel 3) Parallel and are in the same direction 4) Parallel and are in the opposite direction 5. If a b a b .   then the vectors a and b are 1) Like vectors 2) Unlike vectors 3) Equal vectors 4) Perpendicular vectors 6. If a b c , , are three non-zero vectors then a b, = a c, implies that 1) a is orthogonal to (b+c ) 2) a is orthogonal to both b and c 3) b=c+a 4) a is orthogonal to b-c  (or) b c  7. Equality holds in the triangle inequality a+b a + b if  1) a mb  2) a mb  ,m>0 3) a mb  ,m<0 4) 0 ( ) 90 a ,b  8. The vector equation a b a c . .  need not always imply 1) a  0 2) b c  C.U.Q