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Properties of scalar product
Property 1 : The scalar product of two vectors is commutative av.bv = bv.av Property 2 : Scalar Product of Collinear Vectors : (i) When the vectors av and bv are collinear and are in the same direction, then θ = 0 av.bv = |av| |bv| = ab (i) When the vectors av and bv are collinear and are in the opposite direction, then θ = π av.bv = |av| |bv|(-1) = -ab Property 3 : Sign of Dot Product The dot product av.bv may be positive or negative or zero. (i) If the angle between the two vectors is acute (i.e., 0 < θ < 90°) then cos θ is positive. In this case dot product is positive. (ii) If the angle between the two vectors is obtuse (i.e., 90 < θ < 180) then cos θ is negative. In this case dot product is negative. (iii) If the angle between the two vectors is 90° (i.e., θ = 90°) then cos θ = cos 90° = 0. In this case dot product is zero.
scalar product in terms of components If a = a1i+a2j+a3k and b= b1i+b2j+b3k then a.b = a1b1+a2b2+a3b3 Angle between two vectors If θ is the angle between two vectors, cos θ = a.b/|a||b| => θ = cos-1 (a.b/|a||b|) In component form If a = a1i+a2j+a3k and b= b1i+b2j+b3k
θ = cos-1[(a1b1+a2b2+a3b3)/(SQRT(a1²+a2²+a3²)*SQRT(b1²+b2²+b3²))
Components of a vector b along and perpendicular to vector a Component of vector b along vector a == (a.b/|a|²)aComponent of vector b perpendicular to vector a = b- (a.b/|a|²)a
Vector product 26. Definition: a and b are two non-zero non-parallel vectors. Then the vector product a×b is defined as a vector whose magnitude is |a||b| sin θ where θ is the angle between a and b and whose direction is perpendicular to the plane of a and b in such a way that a,b and this direction constitute a right handed system. More about the direction: If η is a unit vector in the direction of a×b, then a,b and η form a system in such a way that , if we rotate vector a into vector b, then η will point in the direction perpendicular to the plane and a and b in which a right handed screw will move if it is turned in the same manner. Magnitude of a×b = |a||b| sin θ Geometrical interpretation of vector product a×b is a vector whose magnitude is equal to the area of the parallelogram having a and b as its adjacent sides. | a×b| = |a | |b| sin θ |a| is the base and |b| sin θ is the height of the parallelogram 27. Properties of vector product a and b are vectors 1. Vector product is not commutative a×b ≠ b×a But a×b = - b×a 2. m is a scalar m a×b = m(a×b) = a×mb 3. m and n are scalars m a×nb = mn a×b = m( a×nb) = n(ma×b) 4. Distributive property over vector addition a×(b+c) = a×b + a×c (left distributivity) (b+c) ×a = b×a + c×a (right distributivity) 5. a×(b-c) = a×b - a×c (left distributivity) (b-c) ×a = b×a - c×a (right distributivity) 6. The vector product of two non-zero vectors is zero is they are parallel or collinear 28. Vector product in terms of components a = a1i+a2j+a3k b = b1i+b2j+b3k a×b = |i j k| |a1 a2 a3| |b1 b2 b3|
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