Vector cross product
Vector cross product is used to calculate area of triangle, rectangle or parallelogram. We can calculate moment of force or torque using cross product of vector.
Vector product or cross product: The vector product of two vectors 
and 
is defined as the product of the magnitudes of and and the sine of the angle between them.
If and creates angle θ then 
= AB
. Where 
is the unit vector perpendicular to the plane of and .
Properties of cross product:
(i) = BA (-) = -(
) and ||= BA
= AB = || [cross product is anticommutative]
(ii) (+
) = + [cross product is distributive]
(iii) If the angle between and is 
then = ABsin = 0
(iv) = A.Asin = 0
(v) If the angle between and is 
then = ABsin = AB.
(vi) 
= 
= 
= 1.1sin = 0
(vii) = , = , = .
Concepts on dot product
1. Area:
Area of a triangle: 
and 
are represented by two sides AB and AC of 
ABC and CD is perpendicular to AB. Angle between and is 
.
The area of ABC = 
CD.AB = ACsin AB = QsinP = PQsin = | 
|.
The direction of area vector is perpendicular outward to the plane of the triangle.
Area of a parallelogram: ABCD is a parallelogram whose sides AB and AD are represented by and .
Area of a parallelogram = 2 Area of a triangle = 2 | | = | |.
Example: 1. ABC is a triangle and the coordinates of vertices A, B and C are respectively (1, 2, 3), (2, -1, 1) and (3, 1, -2). Find the area of ABC.
= 
= (3 -2)î + (1+1)ĵ + (-2-1)k̂ = î + 2ĵ -3k̂and = 
= (1-2)î + (2+1)ĵ + (3-1)k̂ = – î + 3ĵ +2k̂.
The area of ABC = | |
= 
= (4+9)î + (+3-2)ĵ + (3+2)k̂ = 13î + ĵ + 5k̂.
| | = 
= 
.
2. The diagonals of a parallelogram are represented by two vectors 
= 5î – 4ĵ + 3k̂ and 
= 3î + 2ĵ – k̂. Calculate the area of the parallelogram.
Let us consider, and are two sides of the parallelogram. Then, + = and – = .
= ( + ) ( – ) = ( ) – ( ) + ( ) – ( ) = 2( ).
So, area of parallelogram = | | = 
| | = 
= (4 – 6)î + (9 + 5)ĵ + (10 + 12)k̂ = – 2î + 14ĵ +22k̂
So, area = 2
= 
.
2. Torque: Torque or moment of force about point Q = Force at point P Perpendicular distance QO = F QPsin = FRsin = RFsin
= 
. The direction of torque is perpendicular outward to the plane of and .
Example: Force = 2î + 3ĵ – k̂ acts at point P (1, 2, 3). Find torque about point Q (2, 1, 1).
= 
= (1 -2)î + (2-1)ĵ + (3-1)k̂ = -î + ĵ + 2k̂
= = 
= (-1-6)î + (4-1)ĵ + (-3-2)k̂ = -7î + 3ĵ – 5k̂.
3. Perpendicular unit vector: What is the perpendicular unit vector of 
= î +2ĵ – k̂ and 
= 2î + ĵ + 2k̂?
The perpendicular vector of and is 
= | |= 
= (4+1)î + (-2-2)ĵ + (1-4)k̂ = 5î -4ĵ -3k̂.
Unit vector of is 
= 
= 
= 
.
4. Volume of a parallelopiped: The three sides of a parallelepiped are represented by , and . Area of base of parallelepiped (parallelogram) is = | |
where is the unit vector perpendicular to base of parallelepiped.
Height h of the parallelepiped is h = . (projection of on .)
Volume of parallelepiped = height area = .| | = .( ).
Vector triple product: .( ) = .( ) = .( )
If = 
, = 
and = 
then,
.( ) = 
= – 
= 
= .( )
.( ) = = – 
= 
= .( 
)
Therefore, .( ) = .( ) = .( ).
.( ) = = 
+ 
+ 
If , and are coplanar then .( ) = 0.
Example: What is the value of a, for which = (4î –ĵ +3k̂), = (2î +ĵ -2k̂) and = (aî + ĵ – k̂) are the coplanar vectors?
Here .( ) = 0
Or, 
= 0
Or, 4(-1+2) -1(-2a +2) +3(2-a) = 0
Or, 4+2a-2+6-3a = 0
a = 8.
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