10-6:Vectors Product of Two Vectors

Cross Products

• The Cross product of two vectors a x b is perpendicular to the plane containing vectors a and b.
• The magnitude of a x b- the absolute value of vector a multiplied by the absolute value of vector b multiplied by sin(pheta).
  
• The direction of a x b is determined by right hand rule.
  

Right hand rule- Put the fingers of your right hand so they curl in the shortest direction from the first vector to the second vector. The cross product is in the same direction your thumb points.

• Thumb points in-rotates from vector b to a and b x a
  
• Thumb points out-rotates from vector a to b-a x b
  

Cross products of the unit coordinate vector

i x i=0 i x j=k j x i=-k

j x j=0 j x k=i k x j=-i

K x K=0 k x i=j I x k=-j

Example of Cross Product

Find vectors a x b
a= 4i+5j+9k and b=11i+5j+10k
a x b= 44 i x i + 20 i x j + 40 i x k
+ 55 j x i+ 25 j x j + 50 j x K
+ 99 k x i + 45 k x j + 90 k x k

2ok-40j
-55 k + 50 i
99 j - 45 i

=5i+59j-35k

In determinants-Example 1

(i J K )
( 4 5 9 )
(11 5 10)

i(5 9 ) -j (4 9) +K (4 5)
(5 10) (11 10) (11 5)

I(5)(10)-(5)(9) - j(4)(10)-(11)(9)+ k(4)(5)-(11)(5)

=5i+59j-35k

Geometrical cross products and meaning of a x b

area of a parallelogram have a and b as adjacent sides= absolute value (a x b)

area of triangle have a and b as adjacent sides= 1/2 absolute value (a x b)

Example

Find the are of the triangle with vertices's P1(-5, 5, 5), P2(-3, 2, 7), P3(1, 12, 6)

P1P2 x P1P3= -17i + 10 j + 32k

Area= 1/2 (-17i + 10 j + 32k)= 1/2 square root of (-17)^2 + 10^2 + 32^2

= 37.5898


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Additional sources- http://en.wikipedia.org/wiki/Cross_product- This website page gives one great detail on the subject of Cross Products.

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