Last Few Vector Questions

[chee]

Q1: An airplane travels N40∘E at an airspeed of 1000km/h. Measurement on the ground shows that the plane is travelling N45∘E at a speed of 1050 km/h. Calculate the velocity of the wind.

I know we use the formula V_g = V_a + V_w

we know Vg is 1050km/h N45∘E
and Va is 1000km/h N40∘E
we don't know Vw, so we subtract it to the other side of the equation and we have a subtraction of vectors


I drew my vector diagram, and applied to cos law to get the veloctiy of the wind,


a= sqrt[1050^2 + 1000^2 -2(1000)(1050)cos∘]
and got 102.4 km/h as my wind velocity,


can someone confirm if i'm right? :s

Q2: solve for x given vector u=[3x, 7]; vector v= [5x, x]; the magnitude of vector u plus vector v = 10x

i added vector u and v and got [8x, x+7]

then did, 10x= sqrt[(8x)^2 + (x+7)^2] and solved for x

and got x as -1, but when i subbed in -1 and readded vector u and v and found the magnitude i didn't get back -10

 

[pappus]

Q1: An airplane travels N40∘E at an airspeed of 1000km/h. Measurement on the ground shows that the plane is travelling N45∘E at a speed of 1050 km/h. Calculate the velocity of the wind.

I know we use the formula V_g = V_a + V_w

we know Vg is 1050km/h N45∘E
and Va is 1000km/h N40∘E
we don't know Vw, so we subtract it to the other side of the equation and we have a subtraction of vectors


I drew my vector diagram, and applied to cos law to get the veloctiy of the wind,


a= sqrt[1050^2 + 1000^2 -2(1000)(1050)cos∘]
and got 102.4 km/h as my wind velocity, <-- correct


can someone confirm if i'm right? :s

Q2: solve for x given vector u=[3x, 7]; vector v= [5x, x]; the magnitude of vector u plus vector v = 10x

i added vector u and v and got [8x, x+7]

then did, 10x= sqrt[(8x)^2 + (x+7)^2] and solved for x

and got x as -1, but when i subbed in -1 and readded vector u and v and found the magnitude i didn't get back -10

1. Your result is OK.

2. The two vectors are \(\displaystyle \displaystyle{\vec u = \langle -3,7 \rangle}\) and \(\displaystyle \displaystyle{\vec v = \langle -5, -1 \rangle}\) whose sum is \(\displaystyle \displaystyle{\vec u + \vec v = \langle -8,6 \rangle}\)

3. A magnitude is always positiv! Thus \(\displaystyle \displaystyle{|\langle -8,6 \rangle| = \sqrt{64+36}=10}\)