Determine unit vector which is perpendicular to both A=2i+j+k and B=i-j+2k?

Joyce Decker

Joyce Decker

Answered question

2023-02-28

Determine unit vector which is perpendicular to both A=2i+j+k and B=i-j+2k?

Answer & Explanation

Hunter Hendricks

Hunter Hendricks

Beginner2023-03-01Added 6 answers

We are aware that any two vectors' cross products result in a vector that is perpendicular to both vectors.
#:.# for two vectors A and B if C is the vector perpendicular to both.
C = A × B = [ i ^ j ^ k ^ A 1 A 2 A 3 B 1 B 2 B 3 ]
= ( A 2 B 3 B 2 A 3 ) i ^ ( A 1 B 3 B 1 A 3 ) j ^ + ( A 1 B 2 B 1 A 2 ) k ^
Inserting given vectors we obtain
[ i ^ j ^ k ^ 2 1 1 1 - 1 2 ]
= ( 1 × 2 ( - 1 ) × 1 ) i ^ ( 2 × 2 1 × 1 ) j ^ + ( 2 × ( - 1 ) 1 × 1 ) k ^
= 3 i ^ 3 j ^ 3 k ^
Now unit vector in the direction of C is C | C |
| C | = 3 2 + ( - 3 ) 2 + ( - 3 ) 2
= 27
= 3 3
the desired unit vector is as a result
1 3 ( i ^ j ^ k ^ )

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