VectorsInSpace

# Vector product

Let *u* and *v* linearly independent vectors. How do we determine all the vectors that are orthogonal to both of them?

We can suppose that:

We name this vector

**cross or vector product**of

*u*and

*v*.

Then:

- Its magnitude is:

- Its direction is perpendicular to

*u*and*v*.- Its sense is determined by the “corkscrew rule” or “right-hand rule”

*Properties*:

(vii) The area of the parallelogram formed by

*u*and*v*is the magnitude of its vector product.Demonstration:

**Exercise**: Let *A(1,1,1),B(2,-1,0),C(3,3,-2)*. Calculate:

a) *ABxAC*

b) A unit vector orthogonal to *AB* and *AC*

c) The area of the parallelogram defined by the vectors *AB* and *AC*

d) The area of the triangle *ABC*

Solutions: a) (-4,1,-2); b) (-4√21/21,√21/21,-2√21/21); c) √21u^{2}; d) √21/2u^{2}

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