the vector P=ai+aj+3k and Q=ai-2j-k are perpendicular to each other. the positive value of a is :

* As P and Q are perpendicular to each other their DOT PRODUCT or SCALAR PRODUCT will be o.*

(P).(Q)=(ai+aj+3k).(ai-2j-k)                                                 [I hope you know what is dot product and how to solve it]

=>a^2_2a-3                                                                

=a(a-3)+1(a-3)

=(a+1)(a-3)

=>  a=-1 or 3

POSITIVE VALUE=3

a=-1 or a=3

by using    MOD( Pvector +Q vector ) = √(P² +Q²)

Since both the vectors are perpendicular to each other therefore there dot product will be zero. because cos90=0

now p.q=0

      (ai+aj+3k).(ai-2j-k)=0

      a² - 2a - 3 =0

      a² -3a + a -3=0

      a(a-3)+1(a-3)=0

      (a+1)(a-3)=0

therefore .. a=-1 or a=3

since the vectors are perpendicualr their scalar product is zero.

this gives the quadratic equation a² - 2a - 3 = 0

so the positive value of a is 3.