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 → …

Question

Let A , &#160; B , &#160; C be three points whose position vectors respectively are: a &#8594; = i ^ + 4 j ^ + 3 k ^ b &#8594; = 2 i ^ + &#945; j ^ + 4 k ^ , &#160; &#945; &#8712; R c &#8594; = 3 i ^ - 2 j ^ + 5 k ^ If &#945; is the smallest positive integer for which a &#8594; , &#160; b &#8594; , &#160; c &#8594; are non-collinear, then the length of the median, &#9651; A B C , through A is:

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

Let A ,   B ,   C be three points whose position vectors respectively are: a → = i ^ + 4 j ^…

Solution

Let A , &#160; B , &#160; C be three points whose position vectors respectively are: a &#8594; = i ^ + 4 j ^…

Solution

Let A , B , C be three points whose position vectors respectively are: a → = i ^ + 4 j ^…