vectors
Question:
Find two unit vectors in 2-space that make an angle of $45°$ with $4i + 3j$.
I tried letting the other vector, v = ai +bj.
Then I used the dot product trying to obtain and and b,
$(4i+3j).(ai+bj)=5||v||cos45$
$(4a + 3b) = \frac{5}{\sqrt{2}}||v||\sqrt{a^2+b^2}$
$(16a^2 + 9b^2) = \frac{25}{{2}}(a^2+b^2)$
Once I get here, I'm unsure how to progress. I end up with a multi-variable expression I'm not too sure how to solve.
You know (or should know) the cosines of a variety of angles such as $0$, $\frac{\pi}6$, $\frac{\pi}4$, $\frac{\pi}3$, $\frac{\pi}2$, $\pi$, and so on. (Or, if you prefer degrees, $0°$, $30°$, $45°$, $60°$, $90°$, $180°$, and so on.)
Each those pairs of vectors gives a cosine or its negative in that list. So you can do these problems they way you outline.
Another way, however, is to do this geometrically. Sketch a coordinate system and the two vectors. The angle should then be obvious. (I was able to do each problem in my head using this technique.)
Let us know if you cannot figure out a particular vector.
A word of caution: It's highly possible that something is off here, with the frame-of-reference being not quite what I'm used to. Hopefully the ideas are something you can implement and test with a variety of cases to see if something funny happens, with your particular situation. (If anything, I suspect with angles measured clockwise, we might have that $\varphi$ below comes back as $-\varphi$ or something).
I've uploaded another picture with a useful angle marked. If the coordinates of the blue point are $(x, y)$, then the red angle, called $\varphi$ here, can be computed using the $\operatorname{atan2}$ function; that is, $\varphi = \operatorname{atan2}(x, y)$.
Note that both red angles in the picture are $\varphi$.
In the picture, we can see that the green arc is made up of three nice angles:
$$\text{Green Angle} = \varphi + 90^\circ + A,$$ where $A$ is the known angle labeled $20$ in the picture.
A note on $\operatorname{atan2}$: You want your language to have an $\operatorname{atan2}$ function. This is because you need inverse trig functions to find angles, but trig functions are notorious for having "finicky" inverses: if you don't know exactly what you're doing with an inverse trig function, it's easy to get the wrong result half the time. And thus, $\operatorname{atan2}$ was born, it's designed to be very user-friendly :)
Best Answer
The question asks for two unit vectors, so $\|v\|=1$.
Also, I don't see a connection between the equation $$(4i+3j).(ai+bj)=5||v||cos45$$
and
$$(4a + 3b) = \frac{5}{\sqrt{2}}||v||\sqrt{a^2+b^2}$$
Can you explain how one follows from the other? I need to understand your thinking before I can direct it to the answer.