Find the area of the parallelogram whose adjacent sides are determined by the vector .

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Solution

The given adjacent sides of parallelogram are a → = i ^ − j ^ +3 k ^ and b → =2 i ^ −7 j ^ + k ^ .

Area of parallelogram whose adjacent sides are a → and b → is given by,

| a → × b → |

The cross product of two vectors ( a 1 i ^ + a 2 j ^ + a 3 k ^ ) and ( b 1 i ^ + b 2 j ^ + b 3 k ^ )is given by,

a → × b= → | i ^ j ^ k ^ a 1 a 2 a 3 b 1 b 2 b 3 |

a → × b → =| i ^ j ^ k ^ 1 −1 3 2 −7 1 | = i ^ ( −1+21 )− j ^ ( 1−6 )+ k ^ ( −7+2 ) =20 i ^ +5 j ^ −5 k ^

| a → × b → |= ( 20 ) 2 + ( 5 ) 2 + ( 5 ) 2 = 400+25+25 =15 2

Thus, area of parallelogram with adjacent sides a → = i ^ − j ^ +3 k ^ and b → =2 i ^ −7 j ^ + k ^ is 15 2 .

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