Answer:
The angle between vectors u and v is [tex]\theta\approx36.9^\circ[/tex]
Step-by-step explanation:
Recall:
- Angle between vectors u and v --> [tex]\theta=cos^{-1}(\frac{u\bullet v}{||u||*||v||})[/tex]
- Dot product (three-dimensional) --> [tex]u\bullet v=u_1v_1+u_2v_2+u_3v_3[/tex]
- Magnitude (three-dimensional, vector u) --> [tex]||u||=\sqrt{u_1^2+u_2^2+u_3^2}[/tex]
- Magnitude (three-dimensional, vector v) --> [tex]||v||=\sqrt{v_1^2+v_2^2+v_3^2}[/tex]
Calculation:
- Dot product --> [tex]u\bullet v=(4)(6)+(0)(-3)+(2)(0)=24[/tex]
- Magnitude (u) --> [tex]||u||=\sqrt{4^2+0^2+2^2}=\sqrt{16+0+4}=\sqrt{20}=2\sqrt{5}[/tex]
- Magnitude (v) --> [tex]||v||=\sqrt{6^2+(-3)^2+0^2}=\sqrt{36+9+0}=\sqrt{45}=3\sqrt{5}[/tex]
Plug dot product and magnitudes into formula:
[tex]\theta=cos^{-1}(\frac{u\bullet v}{||u||*||v||})=cos^{-1}(\frac{24}{2\sqrt{5}*3\sqrt{5}})=cos^{-1}(\frac{24}{6*5})=cos^{-1}(\frac{24}{30})\approx36.9^{\circ}[/tex]
Conclusion:
The angle between vectors u and v is [tex]\theta\approx36.9^\circ[/tex]
