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

Gaurav Sharma
19 Points
15 years ago

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

Gaurav Sharma
19 Points
15 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|