angle between vectors question...

[bar0029]

Let    u= i - j - k
and   v= i + aj + k, where a is an element of R.
The angle between the vectors is 60 degrees.
Fine the exact value of 'a'.

u.v = |u|.|v|cos60 = 1-a-1 = -a
|u| = 3^1/2
|v| = (2+a^2)^1/2
-a  = (2+a^2)^1/2 * 3^1/2
therefore, a = +(6)^1/2 or a=-(6)^1/2

my question is, when i sub the positive value of a i get 120 degrees
when i sub the negative value i get 60 degrees
so does a = -(6)^1/2  ??

[moekamo]

-a  = (2+a^2)^1/2 * 3^1/2

this line in your working, the RHS is always greater than 0(has a positive square root multiplied by a positive constant), so for the LHS to be always greater than 0, a must be less than 0, ie a<0, which means your answer for a must be negative, which restricts your answer to only the negative value.

[bar0029]

oh okay.

just curious however, if i say, a must be the negative value because then the angle between the two vectors is 60 degrees, is that correct?

[ulbasour]

There is always an obtuse and acute angle between two vectors, you regard only the acute unless otherwise told.