# A dog searching for a bone walks 3.50 m south…..?

A dog searching for a bone walks 3.50 m south, then 8.20 m at an angle of 30.0 degrees north of east, and finally 15.0 m west. Use graphical techniques to find the dog's resultant displacement vector.

The answer in the back of the book says 7.9 m at 4.3 north of west...

• Okay, this will take some work on your part so that you best understand it.

First draw both axes. Once that is done, draw a vector of (theoretical) length 3.5 along the negative x axes.

Then draw a "8.2" vector from that endpoint all the way up, 60 degrees from that point (90-30: this is because you make a 30 degree angle with the current vertical and you will see that the angle in the triangle we are making is 90 degrees minus that 30).

You should make the 8.2 m vector higher than the starting point of the original 3.5 m vector.

Finally draw a 15 m vector from the endpoint of the 8.2 m vector that goes above the original one, and to the left. This one should be drawn really far.

Now we just solve for the sides. Our first triangle is a 30-60-90, so we must solve for the vertical side (the one with same slope as the 3.5 vector), which is just:

8.2/2 = 4.1 m

Now solve for the opposite side using 4.1(sqrt(3)) = 7.1 m. So now we have enough to solve for the other triangle that the 15m line makes.

the top side is 15 - 7.1 = 7.9 m

while the right side is 4.1 - 3.5 = 0.6 m

The last side will be the resultant displacement:

√(7.9^2) + (0.6^2) = 7.9 m (ANSWER)

To find the direction simply use:

arcTan(y/x)=arcTan(0.6/7.9) = 4.3 degrees north of west (ANSWER 2)

• Let east be +x and north be +y, x = distance * cos Î¸, y = distance * sin Î¸

For 3.5 south, y = -3.5

For 8.2, x = 8.2 * cos 30 and y = 8.2 * sin 30

For 15 west, x = -15

Total x = 8.2 * cos 30 – 15 â -7.9

Total y = -3.5 + 8.2 * sin 30 = 0.6

Magnitude = [(8.2 * cos 30 – 15)^2 + 0.6^2)]^0.5

The magnitude is approximately 7.9 m

Tan Î¸ = y/x = 0.6/-7.9

The angle is approximately 4.3Ë North of west

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