Finding second vector given a vector and the angle between the two

Question

I am given a vector w = (1,1) I need to find a second vector v so that the angle between w and v, $\theta$ = $\pi/3$. I am supposed to use this formula to find the solution ||v|| ||w|| cos($\theta$) = w • v

I am at a point where I even think that this is impossible unless I make an assumption but that is not allowed

Any help is appreciated, thanks.

Answer 1

We need to find $v=(a,b)$ such that

$$\frac{v\cdot w}{|v||w|}=\cos \theta=\frac12 \implies 2(a+b)=\sqrt 2\sqrt{a^2+b^2}$$

$$a^2+4ab+b^2=0 \implies b^2\left(x^2+4x+1\right)=0 \quad x=\frac a b$$

$$\implies x_1=-2+\sqrt 3\quad x_2=-2-\sqrt 3$$

therefore we obtain

• $v_1=(-2+\sqrt 3,1)$
• $v_2=(2+\sqrt 3,-1)$

To check we can see geometrically that since vector $w$ is at $45°$ couterclockwise from $x$ axis the vectors $v_i$ are rotated of $-15°$ and $105°$ with respect to $x$ axis.

Answer 2

If you choose another vector $v= (a,b)$. Then according to the formula, you need \begin{equation} w.v= \Vert w \Vert \Vert v \Vert \cos \frac{\pi}{2} = 0 \end{equation} So \begin{equation} (a,b).(1,1) = 0 \end{equation} So \begin{equation} a + b = 0 \end{equation} i.e. $b = -a$ Hence any vector $v = (a,-a)$ will do. Examples are $v = (1,-1)$, $v =(2,-2)$, $v = (\pi, -\pi)$ .. etc.