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Question

The vectors a=3i^-2j^+2k^ and b=-i^-2k^ are the adjacent sides of a parallelogram. The acture angle between its diagonals is _____________.

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Solution

Given:
vectors a=3i^-2j^+2k^ and b=-i^-2k^ are the adjacent sides of a parallelogram



Let θ be the acute angle between the diagonals.The diagonals of the parallelogram area+b=3i^-2j^+2k^ +-i^-2k^ =2i^-2j^ ...1anda-b=3i^-2j^+2k^ --i^-2k^ =4i^-2j^+4k^ ...2The angle between the diagonals is:cosθ=a+b·a-ba+ba-b =24+-2-2+0422+-2242+-22+42 =8+44+416+4+16 From 1 and 2 =12836 =126×22 =12θ=π4

Hence, the acute angle between its diagonals is π4.


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