the unit vector perpendicular to A=2i+3j+k and B=I+j+k is

Asked by suseelakada123 | 25th Jul, 2021, 03:22: PM

Expert Answer:

Let begin mathsize 14px style A with rightwards arrow on top space equals space 2 space i with hat on top space plus space 3 space j with hat on top space plus space k with hat on top end style   and   begin mathsize 14px style B with rightwards arrow on top space equals space space i with hat on top space plus space space j with hat on top space plus space k with hat on top end style
unit vector perpendicular to begin mathsize 14px style A with rightwards arrow on top end style and begin mathsize 14px style B with rightwards arrow on top end style is detromined by taking cross product of  begin mathsize 14px style A with rightwards arrow on top end style and begin mathsize 14px style B with rightwards arrow on top end style and
normalizing the cross product by dividing its magnitude in order to make unit vector .
 
begin mathsize 14px style A with rightwards arrow on top space cross times space B with rightwards arrow on top space equals space open vertical bar table row cell i with hat on top end cell cell j with hat on top end cell cell k with hat on top end cell row 2 3 1 row 1 1 1 end table close vertical bar space equals space i with hat on top space left parenthesis space 3 space minus 1 space right parenthesis space minus space j with hat on top space left parenthesis space 2 minus 1 right parenthesis space plus space k with hat on top space left parenthesis 2 minus 3 right parenthesis space equals left parenthesis space 2 space i with hat on top space minus space j with hat on top space minus space k with hat on top space right parenthesis end style
begin mathsize 14px style open vertical bar A with rightwards arrow on top space cross times B with rightwards arrow on top space close vertical bar space equals space square root of 2 squared plus 1 squared plus 1 squared space end root space equals space square root of 6 end style
Hence unit vector begin mathsize 14px style n with overbrace on top space end style perpendicular to begin mathsize 14px style A with rightwards arrow on top end style and begin mathsize 14px style B with rightwards arrow on top end style is ,   begin mathsize 14px style n with overbrace on top space end style = (1/√6) ( 2 begin mathsize 14px style i with overbrace on top end style - begin mathsize 14px style j with overbrace on top end style - begin mathsize 14px style k with overbrace on top end style )
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If  begin mathsize 14px style P with rightwards arrow on top plus Q with rightwards arrow on top minus R with rightwards arrow on top space equals 0 end style , then begin mathsize 14px style P with rightwards arrow on top plus Q with rightwards arrow on top space equals space R with rightwards arrow on top space end style
Above vector relation is shown in fig.1. Since begin mathsize 14px style R with rightwards arrow on top end style is sum of begin mathsize 14px style P with rightwards arrow on top end style and begin mathsize 14px style Q with rightwards arrow on top end style , We can form a parallelogram with sides  begin mathsize 14px style P with rightwards arrow on top end style and begin mathsize 14px style Q with rightwards arrow on top end style .
 begin mathsize 14px style R with rightwards arrow on top end style is diagonal of parallelogram.  Since all vectors have equal magnitude , this parallelogram becomes rhombus
and is made of two equilateral triangle.

Hence each angle in the rhombus is 60o as shown in figure.  Angle between begin mathsize 14px style P with rightwards arrow on top end style and begin mathsize 14px style Q with rightwards arrow on top end style is 120o .
In similar manner, we get angle between begin mathsize 14px style P with rightwards arrow on top end style and begin mathsize 14px style S with rightwards arrow on top end style is 60 if   begin mathsize 14px style P with rightwards arrow on top plus Q with rightwards arrow on top minus S with rightwards arrow on top space equals 0 end style , or begin mathsize 14px style P with rightwards arrow on top plus Q with rightwards arrow on top space equals space S with rightwards arrow on top space end style
Hence ratio between angles , θ1 : θ2 = 120 : 60 = 2 : 1

Answered by Thiyagarajan K | 25th Jul, 2021, 04:51: PM

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