Inner product and projections

Find the $s$ that makes the following vectors orthogonal: $(1,1,1)$, $(1,s,1)$.

Find the angle $\theta$ between $(1,2,3)$, $(1,1,1)$.

Given that $\langle \vec{v} , \vec{w} \rangle = 1$, $\langle \vec{v} , \vec{u} \rangle = -1$, and $\lVert \vec{v} \rVert = 3$, compute

1. $\langle 3 \vec{u} , 5 \vec{v} \rangle$
2. $\langle \vec{v} , 2 \vec{w} + 3 \vec{u} \rangle$
3. $\langle \vec{w} + 3 \vec{v}, \vec{v} \rangle$

Suppose $\vec{v} = (1,0,-1)$. Find

1. $\operatorname{proj}_{\vec{v}}\bigl( (0,2,1) \bigr)$
2. $\operatorname{proj}_{\vec{v}}\bigl( (1,0,1) \bigr)$
3. $\operatorname{proj}_{\vec{v}}\bigl( (4,-1,0) \bigr)$

The vectors $(1,1,-1)$, $(2,-1,1)$, $(0,1,1)$ form an orthogonal basis. Represent the following vectors in terms of this basis:

1. $(-1,4,0)$
2. $(4,-1,3)$
3. $(-2,6,-2)$

Let $S$ be the subspace spanned by $(2,-1,1)$, $(2,2,2)$. Find an orthogonal basis of $S$ by the Gram--Schmidt process.

Starting with $(1,1,-1)$, $(2,3,-1)$, $(1,-1,1)$, follow the Gram--Schmidt process to find an orthogonal basis of ${\mathbb{R}}^3$.