Find a vector v for which the parallelogram formed by v and w = <2, 2, -1> has area equal to 6.

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tiburtius | High School Teacher | (Level 2) Educator

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You can calculate this by using cross product.


Let's first calculate cross product



Now we calculate norm of that vactor which needs to be equal to 6.




So any vector `v=(v_1,v_2,v_3)` which satisfies the above equation will form parallelogram of area 6 with vector `w.`  Obviously there are infinitely many solutions. One is e.g. if you put `v_1=v_2=0` , your equation becomes


Hence two solutions (out of infinitely many) are `(0,0,3/sqrt2)` and `(0,0,-3/sqrt2).`