Question

You have found a treasure map that directs you to start at a hollow tree, walk 300 meters directly north, turn and walk 500 meters northeast, and then 400 meters at 60° south of east. Since you have been educated about vectors, you decide to save yourself some walking and go directly to the treasure in a straight line from the hollow tree. How far do you have to go, and in which direction?

Answer 1

Answer:633 m

Step-by-step explanation:

First we have moved 300 m in North

let say it as point a and its vector is
`300\hat{j}`

after that we have moved 500 m northeast

let say it as point b

therefore position of b with respect to a is

r
`_(ba)=500cos(45)\hat{i}+500sin(45)\hat{j}`

Therefore position of b w.r.t to origin is

`r_b=r_a+r_(ba)`

`r_b=300\hat{j}+500cos(45)\hat{i}+500sin(45)\hat{j}`

`r_b=500cos(45)\hat{i}+\left [ 250√(2)+300\right ]\hat{j}`

after this we moved 400 m
`60^(\circ)` south of east i.e.
`60^(\circ)` below from positive x axis

let say it as c

`r_(cb)=400cos(60)\hat{i}-400sin(60)\hat{j}`

`r_c=r_(b)+r_(cb)`

`r_c=500cos(45)\hat{i}+\left [ 250√(2)+300\right ]\hat{j}+400cos(60)\hat{i}-400sin(60)\hat{j}`

`r_c=\left [ 250√(2)+200\right ]\hat{i}+\left [ 250√(2)+300-200√(3)\right ]\hat{j}`

magnitude is
`\sqrt{\left [ 250√(2)+200\right ]^2+\left [ 250√(2)+300-200√(3)\right ]^2}`

=633.052

for direction
`tan\theta =(250√(2)+300-200√(3))/(250√(2)+200)`

`tan\theta =(307.139)/(553.553)`

`\theta =29.021^(\circ)` with x -axis