# How do I do this trigonometry vector calculation?

## Homework Statement

A pilot wishes to fly at maximum speed due north. The plane can fly at 100km/h in still air. A 30km/h wind blows from the south-east.
Calculate:
a) The direction the plane must head to fly north.
b) Its speed relative to the ground.

## Homework Equations

Sine Rule: a/SinA=b/SinB=c/SinC
Cosine Rule: a2=b2+c2-2acCosA

## The Attempt at a Solution

I attached a photo because I didn't know how to do maths on here.
The answer I got was 9.93 degrees and 123km/h

https://pasteboard.co/HwvgTjt.png
The answers from the sheet I'm working off say 12.24 degrees and 119km/h. And I don't understand how their way is correct. To get the angle they did: sin45/100=sinθ/30 If 100 is the planes speed it wouldnt be opposite to the angle of the direction of the wind.

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## Answers and Replies

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Delta2
Homework Helper
Gold Member
Your triangle scheme is not correct. According to your scheme, the plane final speed (which is represented by the side b=(AC)) isn't towards north but somewhere between North-Northeast. Make again the triangle in such a way that the side b is vertical towards north and has magnitude unknown which we wish to find, side a is the direction of wind (from south east to northwest, makes 45 degrees angle with side b) and side c has magnitude 100 and direction unknow theta which we wish to find.

NascentOxygen
Staff Emeritus
It is useful to memorise a vector equation that your relative motion triangles must always obey:

$v_b\ =\ v_a\ +\ v_{b\ rel\ a}$

and remember that to show this vector addition, the two vectors being summed need to be drawn head-to-tail.

You are making things difficult for yourself if you don't draw North as heading vertically up the page.

Stephen Tashi