Let A , B , C be three points whose position vectors respectively are: a → = i ^ + 4 j ^ + 3 k ^ b → = 2 i ^ + α j ^ + 4 k ^ , α ∈ R c → = 3 i ^ - 2 j ^ + 5 k ^ If α is the smallest positive integer f

Question

Let A , B , C be three points whose position vectors respectively are: a → = i ^ + 4 j ^ + 3 k ^ b → = 2 i ^ + α j ^ + 4 k ^ , α ∈ R c → = 3 i ^ - 2 j ^ + 5 k ^ If α is the smallest positive integer for which a → , b → , c → are non-collinear, then the length of the median, △ A B C , through A is:

TestHub · Accepted Answer

Let us assume that vectors are collinear so, $\vec{A B} ∥ \vec{A C}$ if $\frac{1}{2} = \frac{α - 4}{- 6} = \frac{1}{2} \Rightarrow α = 1$

Now, for $\vec{a}, \vec{b}, \vec{c}$ to be non-collinear smallest positive integer will be $α = 2$

So, mid-point of $B C = M (\frac{5}{2}, 0, \frac{9}{2})$

Now, $A M = \sqrt{\frac{9}{4} + 16 + \frac{9}{4}} = \frac{\sqrt{82}}{2}$

Mathematics - Vector Question with Solution | TestHub

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