SOLUTION: Find the lengths of both altitudes in the parallelogram determined by [2, 3] and [−5, 7].

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Question 1154874: Find the lengths of both altitudes in the parallelogram determined by [2, 3] and [−5, 7].
Answer by ikleyn(52780) About Me  (Show Source):
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The area of the parallelogram is the absolute value of this determinant comprised of the given vectors' components 


   det %28matrix%282%2C2%2C++2%2C3%2C++-5%2C7%29%29 = 2*7 - 3*(-5) = 14 + 15 = 29.


So, the area of the parallelogram is 29 square units.


The length of the vector [2,3] is  sqrt%282%5E2%2B3%5E2%29 = sqrt%284+%2B+9%29 = sqrt%2813%29.


Hence, the height of the parallelogram drawn to this side  [2,3]  is  29%2Fsqrt%2813%29.


The length of the vector [-5,7] is  sqrt%28%28-5%29%5E2%2B7%5E2%29 = sqrt%2825+%2B+49%29 = sqrt%2874%29.


Hence, the height of the parallelogram drawn to this side  [-5,7]  is  29%2Fsqrt%2874%29.

Solved, answered, explained and calculated.