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Page 1 Page # 32 VECTORS ?. Position Vector Of A Point: let O be a fixed origin, then the position vector of a point P is the vector OP . If a ? and b ? are position vectors of two points A and B, then, AB = a b ? ? ? = pv of B ? pv of A. A. DISTANCE FORMULA : Distance between the two points A ) a ( ? and B ) b ( ? is AB = b a ? ? ? SECTION FORMULA : n m b m a n r ? ? ? ? ? ? . Mid point of AB = 2 b a ? ? ? . ??. Scalar Product Of Two Vectors: a . b ? =  a ?   b ?  cos ?, where  a ? ,  b ?  are magnitude of a ? and b ? respectively and ? is angle between a ? and b ? . 1. i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 ? projection of  b  a.b a on b ? ? ? ? ? ? 2. If ? a = a 1 i + a 2 j + a 3 k & b ? = b 1 i + b 2 j + b 3 k then a.b ? ? = a 1 b 1 + a 2 b 2 + a 3 b 3 3. The angle ? between a&b ? ? is given by a  b a.b cos ? ? ? ? ? ? , 0 ?? ?? ?? ? 4. a . b 0 a b ? ? ? ? ? ? ? (a ? 0 b ? 0) ? ? Page 2 Page # 32 VECTORS ?. Position Vector Of A Point: let O be a fixed origin, then the position vector of a point P is the vector OP . If a ? and b ? are position vectors of two points A and B, then, AB = a b ? ? ? = pv of B ? pv of A. A. DISTANCE FORMULA : Distance between the two points A ) a ( ? and B ) b ( ? is AB = b a ? ? ? SECTION FORMULA : n m b m a n r ? ? ? ? ? ? . Mid point of AB = 2 b a ? ? ? . ??. Scalar Product Of Two Vectors: a . b ? =  a ?   b ?  cos ?, where  a ? ,  b ?  are magnitude of a ? and b ? respectively and ? is angle between a ? and b ? . 1. i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 ? projection of  b  a.b a on b ? ? ? ? ? ? 2. If ? a = a 1 i + a 2 j + a 3 k & b ? = b 1 i + b 2 j + b 3 k then a.b ? ? = a 1 b 1 + a 2 b 2 + a 3 b 3 3. The angle ? between a&b ? ? is given by a  b a.b cos ? ? ? ? ? ? , 0 ?? ?? ?? ? 4. a . b 0 a b ? ? ? ? ? ? ? (a ? 0 b ? 0) ? ? Page # 33 ?? ?. Vector Product Of Two Vectors: 1. If ? ? a b & are two vectors & ? is the angle between them then n sin b a b x a ? ? ? ? ? ? ? , where ? n is the unit vector perpendicular to both ? ? a b & such that n & b , a ? ? ? forms a right handed screw system. 2. Geometrically b x a ? ? = area of the parallelogram whose two adjacent sides are represented by b & a ? ? . 3. 0 k ˆ k ˆ j ˆ j ˆ i ˆ i ˆ ? ? ? ? ? ? ? ; j ˆ i ˆ k ˆ , i ˆ k ˆ j ˆ , k ˆ j ˆ i ˆ ?? ? ? ?? 4. If ? a = a 1 i ˆ +a 2 j ˆ + a 3 k ˆ & ? b = b 1 i ˆ + b 2 j ˆ + b 3 k ˆ then 3 2 1 3 2 1 b b b a a a k ˆ j ˆ i ˆ b a ? ? ? ? 5. b and a o b a ? ? ? ? ? ? ? ? are parallel (collinear) )0b,0a( ? ? ? ? i.e. b K a ? ? ? , where K is a scalar. . 6. Unit vector perpendicular to the plane of a x b a x b n ˆ a & b is ? ? ? ? ? ? ? ? ? If a,b&c ? ? ? are the pv’s of 3 points A, B & C then the vector area of triangle ABC = ? b xb c cxa ? xa 2 1 ? ? ? ? ? ? ? ? . The points A, B & C are collinear if bx bxc c ax 0 a ? ? ? ? ? ? ? ??? ? Area of any quadrilateral whose diagonal vectors are ? ? d 1 & d 2 is given by 1 2 d xd 2 1 ? ? ? Lagrange's Identity : .ab .bb aa. ab. (ax )b a b (a. )b 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Page 3 Page # 32 VECTORS ?. Position Vector Of A Point: let O be a fixed origin, then the position vector of a point P is the vector OP . If a ? and b ? are position vectors of two points A and B, then, AB = a b ? ? ? = pv of B ? pv of A. A. DISTANCE FORMULA : Distance between the two points A ) a ( ? and B ) b ( ? is AB = b a ? ? ? SECTION FORMULA : n m b m a n r ? ? ? ? ? ? . Mid point of AB = 2 b a ? ? ? . ??. Scalar Product Of Two Vectors: a . b ? =  a ?   b ?  cos ?, where  a ? ,  b ?  are magnitude of a ? and b ? respectively and ? is angle between a ? and b ? . 1. i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 ? projection of  b  a.b a on b ? ? ? ? ? ? 2. If ? a = a 1 i + a 2 j + a 3 k & b ? = b 1 i + b 2 j + b 3 k then a.b ? ? = a 1 b 1 + a 2 b 2 + a 3 b 3 3. The angle ? between a&b ? ? is given by a  b a.b cos ? ? ? ? ? ? , 0 ?? ?? ?? ? 4. a . b 0 a b ? ? ? ? ? ? ? (a ? 0 b ? 0) ? ? Page # 33 ?? ?. Vector Product Of Two Vectors: 1. If ? ? a b & are two vectors & ? is the angle between them then n sin b a b x a ? ? ? ? ? ? ? , where ? n is the unit vector perpendicular to both ? ? a b & such that n & b , a ? ? ? forms a right handed screw system. 2. Geometrically b x a ? ? = area of the parallelogram whose two adjacent sides are represented by b & a ? ? . 3. 0 k ˆ k ˆ j ˆ j ˆ i ˆ i ˆ ? ? ? ? ? ? ? ; j ˆ i ˆ k ˆ , i ˆ k ˆ j ˆ , k ˆ j ˆ i ˆ ?? ? ? ?? 4. If ? a = a 1 i ˆ +a 2 j ˆ + a 3 k ˆ & ? b = b 1 i ˆ + b 2 j ˆ + b 3 k ˆ then 3 2 1 3 2 1 b b b a a a k ˆ j ˆ i ˆ b a ? ? ? ? 5. b and a o b a ? ? ? ? ? ? ? ? are parallel (collinear) )0b,0a( ? ? ? ? i.e. b K a ? ? ? , where K is a scalar. . 6. Unit vector perpendicular to the plane of a x b a x b n ˆ a & b is ? ? ? ? ? ? ? ? ? If a,b&c ? ? ? are the pv’s of 3 points A, B & C then the vector area of triangle ABC = ? b xb c cxa ? xa 2 1 ? ? ? ? ? ? ? ? . The points A, B & C are collinear if bx bxc c ax 0 a ? ? ? ? ? ? ? ??? ? Area of any quadrilateral whose diagonal vectors are ? ? d 1 & d 2 is given by 1 2 d xd 2 1 ? ? ? Lagrange's Identity : .ab .bb aa. ab. (ax )b a b (a. )b 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Page # 34 ?V. Scalar Triple Product: ? The scalar triple product of three vectors ? ? ? a b c , & is defined as: ? ? ? ? ? ? a x b c a b c . ? sin cos ? ? . ? Volume of tetrahydron ] c b a [ V ? ? ? ? ? In a scalar triple product the position of dot & cross can be interchanged i.e. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? abxc axbcORabc bca cab .( )( ). [] [] [] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a bx c a cx b i e a b c a c b . ( ) .( ) . . [ ] [ ] ? ? ? ? ? If ? a = a 1 i+a 2 j+a 3 k; ? b = b 1 i+b 2 j+b 3 k & ? c = c 1 i+c 2 j+c 3 k then 3 2 1 3 2 1 3 2 1 c c c b b b a a a ] c b a [ ? ? ? ? . ? If ? ? ? a b c , , are coplanar 0 ] c b a [ ? ? ? ? ? . ? Volume of tetrahedron OABC with O as origin & A( a ? ), B( b ? ) and C( c ? ) be the vertices = ] c b a [ 6 1 ? ? ? ? if the pv’s of its vertices are ? ? ? ? a , b , c & d are given by 1 4 [ ] ? ? ? ? a ? b ? c ? d . V. Vector Triple Product: ? ? ? a x ( b x c) = (. ) (. ) ? ? ? ? ? ? a c b ? a b c , ( ) ? ? ? a x b x c =( .) ( .) ? ? ? ? ? ? a c b ? b c a ? ( ) ( ) ? ? ? ? ? ? a xb xc ? a x b xc , in generalRead More
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