Vector graph please help!?

Question

hi, i got a grade 10 science test tomorrow and for those whoo done vector graphs before i would like to know how to find a propery scale. I have uploaded a screen shot of the work with the asnwers on it but i want to know how they got that scale. (i circled it in red)please check the link http://img443.imageshack.us/my.php?image=untitleddsadasdlh0.png thanks! =)

srry if the screen shots a bit blurry the question says An elephant walks 25 km [N], 12 km [W], 32 km [S], and 21 km [E]. Use
a vector diagram to find the elephant’s displacement.

Answer 1

The scale is just how they drew the graph - 1 cm is supposed to represent 2 km. You'd have to assume, then, that 2 blocks is 1 cm - I counted the blocks, and they all match up to what the problem was saying (25 km north = 25 blocks, 12 km west = 12 blocks, etc.). It's a bit weird because you'd naturally think that each block is 1 cm, but really it's 0.5 cm (and even then, it depends on if the photocopy is accurate - each block may be slightly more or less than 0.5 cm depending on the copier, the original copy, etc.). But in any event, the scale is really determined by the person that drew the graph/chart.

As for solving the problem, you can kinda figure it out if you can see the triangle involved - the elephant is 9 km east of where he started (one leg of a right triangle) and 7 km south of where he started (the other leg). Using the Pythagorean theorem, you can find that the elephant walked:

c^2 = a^2 + b^2
c^2 = 9^2 + 7^2
c^2 = 81 + 49
c^2 = 130
c ~ 11.4 km miles

As for the angle, that's SOH CAH TOA stuff - since you have two legs of the triangle, you can use tan to figure out the angle of the vector from the original point to the final point:

tan (theta) = opp/adj
tan (theta) = 9/7
tan (theta) = 1.29
theta ~ 52 degrees

If you were going to do this problem in i-j terminology (where i represents the x-axis and j represents the y-axis), you'd have:

current position = 25j - 12i - 32j + 21i
current position = 9i - 7j

...which translates into being 9 units right and 7 units down from where he originally was.