vector

Vectors 377
5.17 GEOMETRICAL INTERPRETATION
The scalar triple product a .( b
c ) represents the volume of the parallelopiped
having a , b , c as its co-terminous edges.
a .( b c ) = a .Area of gm OBDC
= Area of gm OBDC × perpendicular distance
A
between the parallel faces OBDC and AEFG. a
–
= Volume of the parallelopiped
Note. (1) If a .( b
c ) = 0, then a, b,
c are
coplanar.
1
(2) Volume of tetrahedron ( )
6 a b c .
Example 4. Find the volume of parallelopiped if
^
^ ^ ^ ^ ^ ^ ^ ^ ^
a 3 i 7 j 5k, b 3i 7j 3k,
and c 7 i 5 j 3k
are the three co-terminous edges of the parallelopiped.
Solution.
Volume = a .( b
c )
3 7 5
=
3 7 3
7 5 3
= 108 – 210 – 170 = – 272
Volume = 272 cube units.
= – 3 (–21 – 15) – 7 (9 + 21) + 5 (15 – 49)
Example 5. Show that the volume of the tetrahedron having
Ans.
A B, B C,
C A as
concurrent edges is twice the volume of the tetrahendron having A , B , C
as concurrent edges.
1
Solution. Volume of tetrahendron = ( ) .[( ) ( )]
6 A B B C C A
1
= ( ) .[ ]
6 A B B C B A C C C A
[ C C 0]
1
= ( ) .( )
6 A B B C B A C A
1
= [ .( ) .( ) .( ) .( ) .( ) .( )]
6 A B C A B A A C A B B C B B A B C A
1 1
= [ A.( BC) B.( CA)] A.( B
C)
6 3
= 2 1 [ ]
6 ABC
= 2 Volume of tetrahedron having A , B , C
, as concurrent edges. Proved.
EXERCISE 5.1
1. Find the volume of the parallelopiped with adjacent sides.
OA = 3 i j, OB j 2 k, and OC i 5 j 4k
extending from the origin of co-ordinates O. Ans. 20
2. Find the volume of the tetrahedron whose vertices are the points A (2, –1, –3), B (4, 1, 3)
C (3, 2, –1) and D (1, 4, 2).
O
n^
–
b
G
B
–
c
C
E
Ans.
F
D
1
7 3