Question

(b) Find the angle between u and v to the nearest degree.

Answer 1

Question 1 (b)

$u \cdot v=11\\\\|u|=\sqrt{3^{2}+(-1)^{2}}=\sqrt{10}\\\\|v|=\sqrt{4^{2}+1^{2}}=\sqrt{17}$

So,

$\cos \theta=\frac{11}{\sqrt{10}\sqrt{17}}\\\\\theta=\cos^{-1} \left(\frac{11}{\sqrt{10}\sqrt{17}} \right) \approx \boxed{32^{\circ}}$

Question 2 (b)

$u \cdot v=22\\\\|u|=\sqrt{5^{2}+2^{2}}=\sqrt{29}\\\\|v|=\sqrt{4^{2}+1^{2}}=\sqrt{17}$

So,

$\cos \theta=\frac{22}{\sqrt{29}\sqrt{17}}\\\\\theta=\cos^{-1} \left(\frac{22}{\sqrt{29}\sqrt{17}} \right) \approx \boxed{8^{\circ}}$