the angle between two vectors 4i+7j+6k and 3i+3j-(c/10)k is 60 degree. find 3c/17
Answers
Explanation:
Whenever we have 2 given vectors and we have to find the angle between them, we use dot product . I will be explaining it first and then we can move on to your question.
Dot product is simply one of the two basic modes of vector multiplication. Now suppose you have been given 2 vectors A and B. Their dot product will be given by:-
A.B=|A|*|B|*cos(Y) - - - - - - - - - - - - - - - - - - - - - - - - - (i)
where Y is the angle between them, and |A|,|B| are the magnitudes of the vectors A and B respectively. Don’t ask me why ‘this’ expression because I really don’t know. My teachers told me it was just formulated without any theoretical proof(hope that’s wrong because I love proofs, helps me organise things a little bit more)
Now remember this: If you are given 2 vectors A = ai + bj + ck and B = di + ej + fk then dot product is also obtained by multiplying the corresponding components of the 2 vectors like this :
A.B = a*d + b*e + c*f - - - - - - - - - - - - - - - - - - - - - - - - (ii)
Note that dot product is a scalar product, i.e. multiplying 2 vectors using the dot product will NOT give a vector but a scalar, i.e. a number.
Now I am assuming that you know what is meant by the magnitudes of 2 vectors, but if not here it goes:-
let A = ai + bj + ck. Then |A|= {a^(2) + b^(2) +c^(2)}^(1/2)
Now that we have covered the basics lets move on to your question.
Let us first find the magnitudes of the given 2 vectors. I am naming them A and B.
|A| = { 2^(2) + 3^(2) + 1^(2) }^(1/2) = {4+9+1}^(1/2) = 14^(1/2)
|B| = { (-3)^(2) + 0^(2) + 6^(2) } ^(1/2) = {9+0+36}^(1/2) = 45^(1/2)
using (ii) A.B = 2*(-3) + 3*0 + 1*6 = (-6) + 6 = 0 - - - - - - - - (iii)
using (i) A.B = 14^(1/2) * 45 * (1/2) * cos(Y) - - - - - - - - - - - - (iv)
by (iii) and (iv) :
0 = 14^(1/2) * 45*(1/2) * cos(Y)
cos(Y) = 0
Y = 90 degrees
So the answer to your question is 90 degrees.
You can note that we didn’t need to find the magnitudes of 2 vectors in these 2 vectors so you can remember one more thing. First find dot products using (ii). If they come out to be zero, then no need to find the magnitudes of the vectors but find them if it isn’t the case.
Hope that helps. Please ask me if you have any more problems.