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				   Hi, please give me list of vector product properties..
				   

7 years ago

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Answers : (2)

										

 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

7 years ago
										

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|

7 years ago

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