Answers:
a) [tex]\hat F=(0.83,-0.55) N[/tex]
b) [tex]\hat D=(-0.44,-0.89) m[/tex]
c) [tex]\hat V=(-0.47,0.88) m/s[/tex]
Explanation:
A unit vector is a vector whose magnitude (length) is equal to 1. This kind of vector is identified as [tex]\hat v[/tex] and the way to calculate is as follows:
[tex]\hat v=\frac{\vec v}{|v|}[/tex]
Where:
[tex]\vec v=(x,y)[/tex] is the vector
[tex]|v|=\sqrt{x^{2}+y^{2}}[/tex] is the magnitude of the vector
Having this information clarified, let's begin with the answers:
a) Force Vector
[tex]\vec F=(9.0 \hat i - 6.0 \hat j) N[/tex]
Magnitude of [tex]\vec F[/tex]:
[tex]|F|=\sqrt{(9.0 \hat i)^{2}+(-6.0 \hat j)^{2 }}N=10.81 N[/tex]
Unit vector:
[tex]\hat F=\frac{\vec F}{|F|}[/tex]
[tex]\hat F=\frac{(9.0 \hat i - 6.0 \hat j) N}{10.81 N}[/tex]
[tex]\hat F=\frac{9.0}{10.81} N-\frac{6.0}{10.81}N[/tex]
[tex]\hat F=(0.83,-0.55) N[/tex]
b) Displacement Vector
[tex]\vec D=(-4.0 \hat i - 8.0 \hat j) m[/tex]
Magnitude of [tex]\vec D[/tex]:
[tex]|D|=\sqrt{(-4.0 \hat i)^{2}+(-8.0 \hat j)^{2 }}m=8.94 m[/tex]
Unit vector:
[tex]\hat D=\frac{\vec D}{|D|}[/tex]
[tex]\hat D=\frac{(-4.0 \hat i - 8.0 \hat j) m}{8.94 m}[/tex]
[tex]\hat D=\frac{-4.0}{8.94} Nm+\frac{-8.0}{8.94}m[/tex]
[tex]\hat D=(-0.44,-0.89) m[/tex]
c) Velocity Vector
[tex]\vec V=(-3.50 \hat i + 6.50 \hat j) m/s[/tex]
Magnitude of [tex]\vec V[/tex]:
[tex]|V|=\sqrt{(-3.50 \hat i)^{2}+(6.50 \hat j)^{2}}m/s=7.38 m/s[/tex]
Unit vector:
[tex]\hat V=\frac{\vec V}{|V|}[/tex]
[tex]\hat V=\frac{(-3.50 \hat i +6.50 \hat j) m/s}{7.38 m/s}[/tex]
[tex]\hat V=\frac{-3.50}{7.38} m/s+\frac{6.50}{7.38}m/s[/tex]
[tex]\hat V=(-0.47,0.88) m/s[/tex]
