*On a particular day the velocity of the wind, in terms of and , is , where .*

* and are unit vectors in the directions East and North respectively.*

*To a man travelling due East the wind appears to come from a directionÂ **North West where .*

*When he travels due North at the same speed as before, the wind appears to come from a direction North West where .*

*Find the actual direction of the wind.*

### Solution:

We start by writing

.

We have two equations.

Firstly, when the man travels due East, for some constant . We have and so

.

We know that this, with respect to the man, is coming from North West. This means we have:

And furthermore:

.

Now consider when the man travels due North, . We have still and so

.

We know that this, with respect to the man, is coming from North West. This means we have:

And furthermore:

,

but we have and so

,

so that .

This means that the velocity of the wind looks like:

That is the wind comes from North West. We have that , so the answer to the question asked is N W.

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