# A car is driven 115 km west and then 75 km southwest. What is the displacement of the car from the point of?

... origin (magnitude and direction)?

. km

° south of west

Relevance

Now if we draw our graph with positive y direction = due north = 90° and positive x direction = due east = 0°.

I'm going to ignore the 115km right now and try to resolve the 75 km into how far west and how far south.

Let's have a stright forward right angled triangle with the hypotoneuse = 75, the distance of the base is equal to the distance of the height. Actually if you want to do the trig you can do base = 75cos45, height = 75 sin 45.

both = 75/sqrt(2)

So the position along the x axis is - (115+(75/√2))

The poisition along the y axis (north positive) = -75/√2

Now we just need to convert -(115+(75/√2)),-75/√2 from rectangular (cartesian) to polar coordinates.

Total displacement = √((- (115+(75/√2)))^2+(-75/√2)^2) = approx. 176.2 km

Absolute distance along the y axis is 75/√2

Absolute distance along the x axis is 115+(75/√2)

arc tan (y/x) = 17.52 degrees south of west (which is what you asked for). Now we needed to convert this to the third quadrant by adding 180 degrees. That is 197.52 degrees if we were plotting it on a conventional graph assuming positve x axis is zero degrees and if we were driving a real car using a real compas assuming north is zero degrees with due east being 90 degrees, then we would have a displacement of 176.2km with a compas heading of 270-17.52 = 252.48°

However the actual answer to your question is 176.2 km 17.52° south of west.

• After the car drives 115 km west, it travels 75 km southwest. The direction southwest is 45 degrees from west (towards the south). If we find the west and south components of this distance in the southwest direction, we can use them to compose a right triangle and solve for the distance and angle with the Pythagorean Theorem. Draw out the path of the car to see why the following relationships are valid. west_component = 75 cos 45 = 53 km south_component = 75 sin 45 = 53 km Total distance to the west = 115 + 53 = 168 km Total distance to the south = 53 km Use the equation c = sqrt(a^2 + b^2) to find c = 176 km, the magnitude of the displacement. Use either the sine, cosine or tangent function to find that the angle of the displacement = 17.5 degrees south of west.