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Normalize the normal vector $(2,4,3)$, whose Euclidean norm is $\sqrt{29}$. The plane has the formThe scalar $d$ is the signed perpendicular distance from the origin in the chosen unit normal direction; the unsigned distance is $|d|$. The nearest point is $d\mathbf{n}$.

Subtracting the first equation from the third gives $x-z=3/2$. The second equation then becomes $(\alpha -2)x=\beta -3$, and the third fixes $y=(7-7x)/5$. The linear system has the following three consistency cases:The first case is a unique solution, the second is an affine line in a vector space of solutions, and the third is an inconsistent linear system.

The common real variable gives the parametric equation of the line:Write $p=(2,-1,4)$ and $u=(1,3,2)$. The point closest to the origin is orthogonal to $u$, so $t_{\ast }=-p\cdot u/|u{|}^2=-7/14=-1/2$. Its position is $(3/2,-5/2,3)$. The distance from a point to a line in three dimensions is therefore

The characteristic polynomial of the Hermitian matrix isSolving the two eigenvector equations and using the Hermitian inner product to normalize givesIndeed $v_1^{†}{v}_2=0$ and both norms equal one. Multiplication of either eigenvector by any unit phase factor gives an equally valid normalized choice.

Where $y\ne 0$ and $x^2\ne 1$, the separable differential equation givesThe requested point selects the implicit solution $2y^2+1=(x^2-1{)}^2$. The two real branches approaching that point areThere is an important initial value problem qualification: the differential equation as printed has no classical solution through the specified initial point. Its right side is undefined at $y=0$, and these branches have an infinite one-sided derivative as $x\downarrow \sqrt{2}$. They satisfy the differential equation on $x>\sqrt{2}$ and extend continuously to the requested point. Multiplying by the vanishing denominator produces an implicit differential equation with a larger domain; it does not repair the original initial value problem.

Expand the determinant along its first row:In particular the matrix has an inverse matrix.