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

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

$r_b=r_a+r_{ba}$

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

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

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

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

$r_c=r_{b}+r_{cb}$

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

$r_c=\left [ 250\sqrt{2}+200\right ]\hat{i}+\left [ 250\sqrt{2}+300-200\sqrt{3}\right ]\hat{j}$

magnitude is $\sqrt{\left [ 250\sqrt{2}+200\right ]^2+\left [ 250\sqrt{2}+300-200\sqrt{3}\right ]^2}$

=633.052

for direction$tan\theta =\frac{250\sqrt{2}+300-200\sqrt{3}}{250\sqrt{2}+200}$

$tan\theta =\frac{307.139}{553.553}$

$\theta =29.021^{\circ}$ with x -axis