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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