Line–plane perpendicularity criterion

Place the intersection point $O$ at the origin, and let $\vec{u}$ and $\vec{v}$ be direction vectors of $a$ and $b$. Since $a$ and $b$ intersect and are distinct lines, $\vec{u}$ and $\vec{v}$ are linearly independent, so together they span every direction lying in the plane $(\alpha)$: the direction vector of any line $c \subset (\alpha)$ can be written as $\vec{w} = \lambda \vec{u} + \mu \vec{v}$ for some real numbers $\lambda, \mu$.

Let $\vec{n}$ be a direction vector of $d$. The hypotheses $d \perp a$ and $d \perp b$ translate to the dot-product equations $\vec{n} \cdot \vec{u} = 0$ and $\vec{n} \cdot \vec{v} = 0$. For a line $c$ in $(\alpha)$ with direction $\vec{w} = \lambda \vec{u} + \mu \vec{v}$, linearity of the dot product gives $\vec{n} \cdot \vec{w} = \lambda(\vec{n}\cdot\vec{u}) + \mu(\vec{n}\cdot\vec{v}) = \lambda \cdot 0 + \mu \cdot 0 = 0$.

Since the dot product of $\vec{n}$ with the direction of an arbitrary line $c \subset (\alpha)$ is zero, $d$ is perpendicular to every line of $(\alpha)$, which is exactly the definition of $d \perp (\alpha)$.