You can
put this solution on YOUR website!I'll do the first part to get you started.
First, multiply
and
to get:
(which is what you have)
Rearrange and group like terms
So we essentially have two complex numbers
and
If we plot these numbers on the complex plane, then we'll have the points (a,b) and (a-b,a+b)
Now think of these points as vectors
So we have two vectors
and
So the goal now is to find the angle between the two vectors (so we'll know the exact angle needed to rotate)
Remember, the angle between two vectors can be found through the formula
=\frac{\v{a}\cdot\v{b}}{\left|\v{a}\right|\left|\v{b}\right|})
where
is the magnitude (ie length) of vector
and
is the magnitude (ie length) of vector
Note: I'm making assumptions that you are familiar with vector calculations (such as the dot product and finding the magnitude of a vector). If my assumptions are wrong, please let me know.
So let's calculate the dot product:
(a-b)+(b)(a+b)=a^2-ab+ab+b^2=a^2+b^2)
Now let's calculate the magnitudes:
^2+(a+b)^2}=\sqrt{a^2-2ab+b^2+a^2+2ab+b^2}=\sqrt{2a^2+2b^2}=\sqrt{2(a^2+b^2)})
Now multiply the magnitudes
}=\sqrt{2(a^2+b^2)(a^2+b^2)}=\sqrt{2(a^2+b^2)^2}=\sqrt{2}\cdot\sqrt{(a^2+b^2)^2}=\sqrt{2}(a^2+b^2))
Now plug these values into the given formula above
=\frac{a^2+b^2}{\sqrt{2}(a^2+b^2)})
Cancel out the common terms
=\frac{1}{\sqrt{2}})
Now take the arccosine of both sides (to solve for theta)
})
Evaluate the arccosine of
to get
radians (or 45 degrees). Note: use the unit circle
or
=====================================================
Answer:
So the vector
needs to rotate
radians (or 45 degrees) in order to be parallel to the vector
Now if any of this is not making any sense, simply pick values for "a" and "b" and plug them in. So let's say that we choose
and
. So the first number would be
and the second number would be
Now find the angle between the vectors <2,3> and <-1,5> and you'll find that it will be 45 degrees.
(