### 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.

- 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

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

Kaori, your up next good luck

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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