# Vector expressions - equivalence

• jemerlia

## Homework Statement

A car is traveling at 12ms^-1. To the passenger in the car the wind appears to be blowing at 8.0ms^-1 at right angles to the road. What is the magnitude and direction of the velocity of the wind with respect to the ground.

## Homework Equations

I can think of two possible vector expressions which should be equivalent. I am clearly doing something wrong because they are not.
Using:
vwg = velocity of wind wrt ground
vcg = velocity of car wrt ground
vwc = velocity of wind wrt car
Expression 1
vwg = vwc + vcg

Expression 2 (following the rule of subtracting the observer's movement)
vwc= vwg-vcg

Yes - I'm aware I should add the negative vector and that the negative sign means the reverse direction

## The Attempt at a Solution

Expression 1:
The solution vector diagram is (excuse the dots so I don't lose the spaces):

^----------->vcg
|.....^
|vwc ... /
|... / vwg
|.../
|../
|/
It gives a vwg magnitude of c. 14ms^-1 and a direction of tan^-1(8/12) = c. 34 degrees with the wind coming from behind the car

Expression 2:
The solution vector diagram is:

<----------------- -vcg
.^......^
...\......|
...\ ...| vwc
...vwg..\....|
.....\...|
.....\...|
......\..|
.....\|
Of course the magnitude is the same as in the previous example but the direction is
tan^-1 (8/-12) = -34 degrees.

My concern is that the wind appears to be coming from the front!

Gloom - what have I misunderstood?

## Answers and Replies

Your method is correct. The answer is not exactly 14m/s. In both cases it is only the direction you start with that results in the -ive or +ive sign. Why do you think the wind appears to be coming from the front?

Thank you for your reassurance. I lacked the confidence to be certain the second vector diagram was the equivalent of the first - especially because vector arithmetic is new to me. I would like to ask this naive question - if a, b and c are vector quantities and they are related:
a = b+c
then is it correct to transpose so that:
b= a-c
?

Why did I think the wind was coming from the front? The honest answer is twofold: first: sometimes I have a pathological inability to imagine the behaviour of the real word and, second I had convinced myself the second expression/vector diagram was telling a different story to the first.