vector

376 Vectors
^ ^ ^
^ ^ ^
= (8 3) i ( 4 6) j (1 4) k = 5i 10 j 5k
2 2 2
Area of parallelogram = (5) (10) (5) = 5 6 Ans.
5.14 MOMENT OF A FORCE
O
Let a force F ( PQ ) act at a point P.
Moment of
F about O
= Product of force F and perpendicular
distance (ON. ^
)
= (PQ) (ON)( ^
) = (PQ) (OP) sin (^
) =
M r F
5.15 ANGULAR VELOCITY
OP PQ
Let a rigid body be rotating about the axis OA with the angular
velocity which is a vector and its magnitude is radians per second
and its direction is parallel to the axis of rotation OA.
Let P be any point on the body such that OP = r and
AOP = and AP OA. Let the velocity of P be V.
Let be a unit vector perpendicular to and r .
r = ( r sin ) ^ = ( AP) = (Speed of P) ^
r
N P F
A
Axis
B
V
r
Q
P
= Velocity of P to and r
Hence V =
r
5.16 SCALAR TRIPLE PRODUCT
Let a, b,
c be three vectors then their dot product is written as a .( b
c)or[
a b c ].
If
a =
^
^ ^ ^ ^ ^ ^ ^
^
1 2 3 , 1 2 3 , and 1 2 3
a i a j a k b b i b j b k c c i c j c k
^ ^ ^ ^ ^ ^ ^ ^ ^
1 2 3 1 2 3 1 2 3
a .( b c ) = ( a i a j a k).[( b i b j b k) ( c i c j c k)]
=
^ ^ ^ ^ ^ ^
1 2 3 2 3 3 2 3 1 1 3 1 2
2 1
( a i a j a k).[( bc bc ) i ( b c b c ) j ( bc bc) k]
= a 1
(b 2
c 3
– b 3
c 2
) + a 2
(b 3
c 1
– b 1
c 3
) + a 3
(b 1
c 2
– b 2
c 1
)
=
a a a
1 2 3
b b b
1 2 3
c c c
1 2 3
Similarly, b .( c
a)and c .( a
b ) have the same value.
a .( b c ) = b .( c
a ) = c .( a
b )
The value of the product depends upon the cyclic order of the vector, but is
independent of the position of the dot and cross. These may be interchanged.
The value of the product changes if the order is non-cyclic.
Note.
a ( b . c) and ( a . b)
c are meaningless.
O