Inner product of Brownian motion and unit vector $\langle x, W_t\rangle$

Question

I'm taking a course on stochastic processes, and I've been presented with some interesting questions. Let $W_t = (W_{1, t}, ..., W_{d, t})$ be a d-dimensional Wiener process. Then

1. If $x\in\mathbb{R}^d, ||x|| = 1$, is $\langle x, W_t\rangle$ also a Wiener process?
2. Set $d=2.$ Define $Y_t = (Y_{1, t}, Y_{2, t}) = \left(W_{1, \frac{2t}{3}} - W_{2, \frac{t}{3}}, W_{2, \frac{2t}{3}} + W_{1, \frac{t}{3}}\right)$. Is $Y_{1, t}$ independent of $Y_{2, t}$?
3. Is $\langle x, Y_t \rangle$ a B.M.?

My thoughts are yes to all questions, but I'm afraid I may have committed an error somewhere.

Answer 1

1. For $s < t$, $$\mathbb E\left[e^{iu\left\langle x, W_t\right\rangle}\Big |\mathcal F_s\right]=e^{-\frac12(t-s) u^2\left\|x\right\|^2} = e^{-\frac12(t-s)u^2}$$

2. $Y$ is Gaussian, so use the covariance to check independence: \begin{align} \texttt{Cov}\left[Y_{1,t}, Y_{2,t}\right] &= 0 + \frac{t}3 - \frac{t}3 + 0 = 0 \end{align}

3. Use the same idea as in 1)