Give the components of the vector whose length is 10 and whose direction opposes the direction of [ -4, 3 ]
Can I get some help pls?
$\displaystyle v_1 = <-4, 3>$
$\displaystyle \theta_1 = \arctan \left( \frac{3}{-4} \right) + \pi \approx 2.498 \,rad$
(we add pi because theta is in the 2nd quadrant)
$\displaystyle \theta_2 = \theta_1 + \pi \approx 2.498 + \pi \approx 5.640 \,rad$
We want our new vector to be in the opposite direction, so add pi again.
One could argue that adding pi twice isn't necessary, and finding arctan (3/-4) would give us the angle we're looking for. But I like angles to be in $\displaystyle [0, 2\pi)$.
Our new vector is
$\displaystyle \begin{aligned}
v_2 &= <|v_2|\cos \theta_2, |v_2|\sin \theta_2> \\
v_2 &= <|10|\cos \,5.640, |10|\sin \,5.640> \\
v_2 &= <8, -6>
\end{aligned}$
EDIT: This long-winded way to solve the problem was deliberate.