Question

A hiker walks 4 km north and then 5 km northeast. Draw displacement vectors representing the hiker's trip and draw a vector that represents the hiker's net displacement from the starting point.

Answer 1

`u+v =8.323717394km`

Ф=
`64.86489406`°

Explanation:

`u=4km`

`v=5km`

For easy calculations let's decompose the vector in its rectangular components:

`u_x=u*cos(\alpha )=4000*cos(90)=4000*0=0`

`u_y=u*sin(\alpha )=5000*sin(90)=4000*1=4000`

`v_x=v*cos(\beta )=5000*cos(45)=3535.533906`

`v_y=v*sin(\beta )=5000*sin(45)=3535.533906`

Now lets calculate the resultant vector:

`u+v=R`

`R=R_x+R_y`

`R_x=u_x+v_x=0+3535.533906=3535.533906`

`R_x=u_y+v_y=4000+3535.533906=7535.533906`

Finally let´s calculate the magnitude and direction of R:

║
`R`║=
`\sqrt{(R_x)^(2)+(R_y)^(2)  }=\sqrt{(3535.533906)^(2)+(7535.533906)^(2)  }  =8.323717394km`

Ф=
`arctan((R_y)/(R_x))=arctan((7535.533906)/(3535.533906))=64.86489406`°