If a → = 2 i ^ + k ^ , b → = i ^ + j ^ + k ^ , find the magnitude of a → × b → .

Given vectors: a → = 2 i ^ + 0 j ^ + k ^ and b → = i ^ + j ^ + k ^ Now, a → × b → = i ^ j ^ k ^ 2 0 1 1 1 1 = ( 0 - 1 ) i ^ - ( 2 - 1 ) j ^ + ( 2 - 0 ) k ^ = - i ^ - j ^ + 2 k ^ Therefore, | a → × b → | = ( - 1 ) 2 + ( - 1 ) 2 + 2 2 = 6 .

If a → = 2 i ^ + k ^ , b → = i ^ + j ^ + k ^ , find the magnitude of a → × b → .

Given vectors: a → = 2 i ^ + 0 j ^ + k ^ and b → = i ^ + j ^ + k ^ Now, a → × b → = i ^ j ^ k ^ 2 0 1 1 1 1 = ( 0 - 1 ) i ^ - ( 2 - 1 ) j ^ + ( 2 - 0 ) k ^ = - i ^ - j ^ + 2 k ^ Therefore, | a → × b → | = ( - 1 ) 2 + ( - 1 ) 2 + 2 2 = 6 .

Find a unit vector perpendicular to both the vectors 4 i ^ - j ^ + 3 k ^ and - 2 i ^ + j ^ - 2 k ^

Let the given vectors be a → = 4 i ^ - j ^ + 3 k ^ and b → = - 2 i ^ + j ^ - 2 k ^ ∴ a → × b → = i ^ j ^ k ^ 4 - 1 3 - 2 1 - 2 = ( 2 - 3 ) i ^ - ( - 8 + 6 ) j ^ + ( 4 - 2 ) k ^ = - i ^ + 2 j ^ + 2 k ^ Therefore, | a → × b → | = 1 + 2 2 + 2 2 = 9 = 3 Unit vector perpendicular to a → and b → = ± a → × b → | a → × b → | = ± - i ^ + 2 j ^ + 2 k ^ 3 .

Find a unit vector perpendicular to both the vectors i ^ + 3 j ^ - 2 k ^ and - i ^ + 3 k ^ .

Let the given vectors be a → = i ^ + 3 j ^ - 2 k ^ and b → = - i ^ + 3 k ^ ∴ a → × b → = i ^ j ^ k ^ 1 3 - 2 - 1 0 3 = ( 9 - 0 ) i ^ - ( 3 - 2 ) j ^ + ( 0 + 3 ) k ^ = 9 i ^ - j ^ + 3 k ^ Therefore, | a → × b → | = 9 i ^ - j ^ + 3 k ^ = 9 2 + - 1 2 + 3 2 = 81 + 9 + 1 = 91 Hence, the unit vector perpendicular to a → and b → = ± a → × b → | a → × b → | = ± 1 91 9 i ^ - j ^ + 3 k ^ .

Find a unit vector perpendicular to both the vectors 3 i ^ + j ^ - 2 k ^ and 2 i ^ + 3 j ^ - k ^ .

Let the given vectors be a → = 3 i ^ + j ^ - 2 k ^ and b → = 2 i ^ + 3 j ^ - k ^ ∴ a → × b → = i ^ j ^ k ^ 3 1 - 2 2 3 - 1 = ( - 1 + 6 ) i ^ - ( - 3 + 4 ) j ^ + ( 9 - 2 ) k ^ = 5 i ^ - j ^ + 7 k ^ Therefore, | a → × b → | = 5 i ^ - j ^ + 7 k ^ = 5 2 + - 1 2 + 7 2 = 25 + 1 + 49 = 5 3 Hence, the unit vector perpendicular to a → and b → = ± a → × b → | a → × b → | = ± 1 5 3 5 i ^ - j ^ + 7 k ^ .