How to Correctly Calculate Vector v Given Magnitude and Angle with u?

[Rapier]

Homework Statement

Let u = <3,-1>. Find v such that ||v||=4 and the angle between the vectors is ∏/3.

Homework Equations

u.v = ||u|| ||v|| cos θ

The Attempt at a Solution

<3,-1>.v = (√10)(4) cos ∏/3
<3,-1>.v = 2√10
v = <(2√10)/3, (2√10)/-1>

Can you please confirm that I am doing this correctly, or if not help me through where I am going awry?

 

[SammyS]

Homework Statement

Let u = <3,-1>. Find v such that ||v||=4 and the angle between the vectors is ∏/3.

Homework Equations

u.v = ||u|| ||v|| cos θ

The Attempt at a Solution

<3,-1>.v = (√10)(4) cos ∏/3
<3,-1>.v = 2√10
v = <(2√10)/3, (2√10)/-1>

Can you please confirm that I am doing this correctly, or if not help me through where I am going awry?

You're not doing this correctly. The v you have has a magnitude that's way too big.

Write vector, v, as v = <v_x, v_y>, where v_x² + v_y² = 16 .

Solve this for v_y.​

Your lines:

<3,-1>∙v = (√10)(4) cos ∏/3

<3,-1>∙v = 2√10​

are fine.

But in the next line, it looks like you tried to divide by the vector <3,-1>. There is no such thing as vector division !

Compute <3,-1>∙v = (3)v_x + (-1)v_y.

Of course that gives you: (3)v_x + (-1)v_y = 2√(10)

Solve that equation and the equation, v_x² + v_y² = 16, simultaneously.