I have a question. Lets say you have three distinct, non zero vectors that lie in the same plane. Can I verify the othoginality of the third vector using strictly cross products?(adsbygoogle = window.adsbygoogle || []).push({});

I know I can just do the dot product and if it equals to 0 then its orthoginal to the plane; however, I came across a method online (cannot find the source) where you can strictly use cross products to verify orthoginality. This had me and my professor thinking about whether or not this may be true such as below:

If i were to take the cross product of lets say A X B. Then take the cross product of A X C. If the resultant vectors of both A X B and A X C are equal, doesn't it technically imply that the third vector C is orthogonal?

For real number values: Lets say the 3 vectors were <1,-1,1>,<-2,3,4>,<0,1,6>.

A X B = | 1 -1 1 | = <-7,-6,1>

| -2 3 4 |

A X C = |1 -1 1| = <-7,-6,1>

|0 1 6|

Since both have the same normal vectors that are perpendicular to the plane, we can conclude that both lie within the same plane.

(Sorry if I do not make sense, its rather late. Perhaps ill revise this tomorrow morning)

Best Regards,

Adrian

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# Cross Products Verifying Third Vector Orthoginality?

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