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Which one of the following vectors of magnitude sqrt(51) makes equal angles with three vectors vec(a)=(hat(i)-2hat(j)+2hat(k))/(3), vec(b)=(-4hat(i)-3hat(k))/(5) and vec(c)=hat(j)? |
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Answer» `5hat(i)-hat(j)-5hat(K)` `:.sqrt(x^(2)+y^(2)+z^(2))=sqrt(51)` `rArr x^(2)+y^(2)+z^(2)=51 " " ...(i)` Let `vec(p)` makes equal angle `theta` with and `vec(a), vec(B) and vec(c)`. `:.vec(p).vec(a)=|vec(p)|.|vec(a)|cos theta` `:. cos theta = (vec(p).vec(a))/(|vec(p)||vec(a)|)` Similarly, `cos theta = (vec(p).vec(b))/(|vec(p)||vec(b)|)` and `cos theta = (vec(p).vec(c))/(|vec(p)||vec(c)|)` `:. (vec(p).vec(a))/(|vec(p)||vec(a)|)=(vec(p).vec(b))/(|vec(p)||vec(b)|)=(vec(p).vec(c))/(|vec(p)||vec(c)|)` `rArr ((1)/(3)(x-2y+2z))/(sqrt(x^(2)+y^(2)+z^(2))(1)/(3)sqrt(1+4+4))=((1)/(5)(-4x-3z))/(sqrt(x^(2)+y^(2)+z^(2))(1)/(5)sqrt(16+9))=(y)/(sqrt(x^(2)+y^(2)+z^(2))sqrt(1))` `rArr (x-2y+2z)/(3sqrt(x^(2)+y^(2)+z^(2)))=(-4x-3z)/(5sqrt(x^(2)+y^(2)+z^(2)))=(y)/(sqrt(x^(2)+y^(2)+z^(2)))` `rArr (x-2y+2z)/(3)=(-4x-3z)/(5)=y` `:. 5(x-2y+2z)=-3(4x+3z)=15Y` `:. 5x-10y+10z=15y and -12x-9z=15y` `rArr 5x-25y+10z=0 and -12x-15y-9z=0` `rArr x-5y+2z=0 and 4x+5y+3z=0` `x-5y+2z=0` `4x+5y+3z=0` `|(x,y,z),(1,-5,2),(4,5,3)|=(x)/(-15-10)=(y)/(8-3)=(z)/(5+20)` `(x)/(-25)=(y)/(5)=(z)/(25)` `(x)/(-5)=(y)/(1)=(z)/(5)=k`(let) `:. x=-5k, y=k, z=5k` Now, `x^(2)+y^(2)+z^(2)=51` `:. (-5k)^(2)+k^(2)+(5k)^(2)=51` `rArr 25k^(2)+k^(2)+25k^(2)=51` `rArr 51k^(2)=51` `:. k=pm1` When, k=1, then `x=-5, y=1, z=5` `and vec(p)=5hat(i)+hat(j)+5hat(k)` when k=-1 , then x=5, y=-1, z=-5 `and vec(p)=5hat(i)-hat(j)-5hat(k)` |
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