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Q.1. In a Triangle ABC, the bisector of angle A meets the opposite side at D. Using vectors Prove that BD : DC = c : b.
Q.2. The vector (-1,1,1) bisects the angle between the vectors C(x,y,z) and (3,4,0). determine a unit vector along C(x,y,z).
Q.3. If a, b, c and a', b', c' are reciprocal system of vectors , then prove that:
a' x b' + b' x c' + c' x a' = (a + b+ c)/[a b c]
Q.4.Prove necessary and sufficient condition for
a x (b x c) = (a x b) x c is that (a x c) x b = 0.
*note in question 3,4 consider a vector sign above the letters a,b,c,a',b',c'. since they are vectors and close bracket [ ] implies scalar triple product of vectors. Moreover 'x' implies Cross Product.
(Questions from FIITJEE Package-Vectors Assignment Level 2)
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6 Answers
713.
Let us assume the following system of reciprocal vectors, viz.,
a' = b x c[abc] ; b' = c x a[abc] ; c' = a x b[abc]
Now you just put these into the given relation and using some formulas of dot and cross product, you can see what that is given follows.
714. Just expand both sides using formula for Vector Triple Product i.e.,
a x (b x c) = (a . c)b - (a . b)c
And cancelling the common term and taking all on one side, we have :
(b . a)c - (b . c)a = 0
Which is obviously the necessary result that follows.