I'm doing my vectors part for pre-calculus and I've been struggling to answer this question for hours. Basically, the question goes like this. 6. Based on the vectors listed in question, 4 compare the angles theta (from their cosine) between (a)between vectors $u$ and $v$
The vectors in question 4 are $u = (1,2)$ and $v=(3,4)$.
The answer has to equal to: $0.1799$
This is what I've tried so far:
I used the standard formula for finding out the angle between two vectors: $\arcsin(\frac{11}{\sqrt{1^2+2^2}}*\sqrt{3^2+4^2})$ and I arrive at the answer:$10.30$ which on the first glance makes sense on the graph, but the answer does not match the one on the answer sheet...
The formula you were expected to use is $$\vec{u}\cdot \vec{v} = |u| |v| \cos \theta_{uv}$$
If you plug these vectors into that formula, you get $$\vec{u}\cdot \vec{v} = 11\\ |u| = \sqrt{5} \\ |v| = 5 \\ \cos \theta_{uv} = \frac{11}{5\sqrt{5}} = \frac{11\sqrt{5}}{25} $$ Notice that this cosine comes out to less than $1$; a value of $10.30$ for the cosine of a real angle should make you smell something rotten.
But the answer they were looking for is the angle itself, which is $$\cos^{-1} \frac{11\sqrt{5}}{25} = 0.179853$$
So it looks like you just used your calculator and got a number of degrees when they wanted a number of radians. You did everything else right.