Wednesday, March 28, 2007

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

-55 k + 50 i
99 j - 45 i


In determinants-Example 1

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

(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)


Geometrical cross products and meaning of a x b

rea 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)


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

Kaori, your up next good luck

Additional sources- This website page gives one great detail on the subject of Cross Products.

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Here a video of his career, check it out-


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