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Question

Find a unit vector perpendicular to each of the vector and , where and .

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Solution

The given vectors are a =3 i ^ +2 j ^ +2 k ^ and b = i ^ +2 j ^ 2 k ^

a + b =4 i ^ +4 j ^ and a b =2 i ^ +4 j ^ .

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 | (1)

A unit vector perpendicular to each vector can be found by their cross product.

( a + b )×( a b )=| i ^ j ^ k ^ 4 4 0 2 0 4 | = i ^ ( 16 ) j ^ ( 16 )+ k ^ ( 8 ) (2)

| ( a + b )( a b ) |= ( 16 ) 2 + ( 16 ) 2 + ( 8 ) 2 = 2 2 × 8 2 + 2 2 × 8 2 + 8 2 =8 2 2 + 2 2 +1 =8 9 (3)

Further simplify the equations,

| ( a + b )( a b ) |=8×3 =24

We need to find unit vector perpendicular to both ( a + b )and ( a b ). So, divide (2) and (3),

± i ^ ( 16 ) j ^ ( 16 )+ k ^ ( 8 ) 24 =± 2 3 i ^ 2 3 j ^ 1 3 k ^

Thus, the unit vector perpendicular to each vectors a =3 i ^ +2 j ^ +2 k ^ and b = i ^ +2 j ^ 2 k ^

can be found by their cross product.


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