# Using cross products to find orthogonal vectors and areas

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Find a vector orthogonal to the plane through the points P, Q, and R and find the area of the triangle PQR.

P(1,0,0) Q(0,2,0) R(0,0,3)

Â© BrainMass Inc. brainmass.com December 24, 2021, 7:34 pm ad1c9bdddfhttps://brainmass.com/math/triangles/using-cross-products-orthogonal-vectors-areas-199562

## SOLUTION This solution is **FREE** courtesy of BrainMass!

To find a vector orthogonal to the plane, first find any two direction vectors in the plane, then take their cross product:

The vector PQ is Q-P=(-1,2,0)

The vector PR is R-P=(-1,0,3)

The cross product of these is orthogonal to each vector, and therefore to the plane in which they lie.

(-1,2,0)x(-1,0,3)=(6,3,2)

The area of a triangle with sides given by the vectors a, b and a+b is given by the formula |axb|/2. Applying this formula, using the cross product we already worked out, we see that the area of the triangle PQR is |(6,3,2)|/2=7/2.

Â© BrainMass Inc. brainmass.com December 24, 2021, 7:34 pm ad1c9bdddf>https://brainmass.com/math/triangles/using-cross-products-orthogonal-vectors-areas-199562