Show that $\{t_1v_1, t_2v_2\}$ is an orthogonal set?

Question

Let {$\vec v$₁ $\vec v$₂} be an orthogonal set of vectors in Rⁿ, and let t₁, t₂ be ∈ R.
Show that {t₁$\vec v$₁, t₂$\vec v$₂} is an orthogonal set.

I know that i probably have to use this property $$(c\vec u)⋅\vec v = c (\vec u ⋅ \vec v) = u⋅(c\vec v).$$

I know that multiplying by different scalars will still give me an orthogonal set.

I don't know how to actually prove it equal to zero. Do i just use simply algebra to prove this by setting one side equal to zero?

Answer 1

Let $\{v_1, v_2\}$ be an orthogonal set of vectors in $\mathbb{R}^{n}$.

Then $\langle v_1 , v_2\rangle = 0$.

Claim : $\{t_1 v_1,t_2 v_2\}$ is also an orthogonal set of vectors for any $t_1 , t_2 \in \mathbb{R}$.

Indeed, we have that

$\langle t_1 v_1,t_2 v_2\rangle = t_1 t_2 \langle v_1, v_2\rangle=0$