Given a line, to draw a "perpendicular [to it]" in Minkowski spacetime...

from my answer https://www.physicsforums.com/threa...me-meaning-visualization.1016318/post-6644678 ,
use Minkowski's construction

From Minkowski's "Space and Time"... $$
We decompose any vector, such as that from O to x, y, z, t into four
components x, y, z, t. If the directions of two vectors are, respectively, that
of a radius vector OR from O to one of the surfaces [itex]\mp F=1[/itex], and that of
a tangent RS at the point R on the same surface, the vectors are called
normal to each other. Accordingly,
[tex]c^2tt_1 − xx_1 − yy_1 − zz_1 = 0[/tex]
is the condition for the vectors with components x, y, z, t and [itex]x_1[/itex], [itex]y_1[/itex], [itex]z_1[/itex], and [itex]t_1[/itex] to be normal to each other.

In other words, draw a "circle"
(in Minkowski spacetime, [the future-branch of] a hyperbola--- the locus of events (say) one-tick-later along inertial-worldlines from event O ) centered at the origin.
To construct a "perpendicular" to a line, find the intersection of your line and the "circle".
Construct the tangent at that intersection point.
(Key idea: The tangent is "perpendicular" to the radius.)
Through the origin, draw a parallel to that tangent line.... That is the x'-axis for the observer along the t'-axis.

desmos.com/calculator/dlsljxj8n2
(Try tuning the E-value: E=1 is Minkowski, E=-1 is Euclidean, E=0 is Galilean)

See also my spacetime diagrammer www.desmos.com/calculator/4765550a89
To see a variation of @Ibix 's construction of the "perpendicular",
click on the circle for "----ticks---lightclockDiamonds"... but only for the E=1 case.
Further down, to see "grids"
you can click on the circles for
coordinateSystem BOB (2)
----ticks---A's rotated graph paper
----ticks---B's rotated graph paper

Along the lines of @Ibix 's construction of the "perpendicular",
the essential part of the construction is
1. constructing the light-cones of two events on an inertial worldline, then
2. constructing the hyperplane that contains the intersection of those light-cones
3. constructing the parallel-hyperplane through the origin

Here's an extension of my earlier desmos file
desmos.com/calculator/hfaxw4bgwq

(The light-cone is a hyperbola (hyperboloid) of "radius zero".)

PeroK said:

I'm one of the few students, it seems, that finds spacetime diagrams confusing. The problem is that the diagram you have drawn is in Minkowski spacetime, and is

not Euclidean

. You cannot apply a Euclidean interpretation of lengths and angles.

Pythagoras's theorem does not apply

!

You have to relearn how to interpret the geometry of a Minkowski spacetime diagram. If I'm honest, I never mastered this. For me, the Lorentz Transformation is too complex to visualise and I rely on algebra.

First, I acknowledge what you say about spacetime diagrams being confusing.
So, some of my efforts have tried to develop additional geometric intuition, not found in typical references.

Second, I do wish to point out that a "PHY-101 position-vs-time graph"
is also a diagram with a non-euclidean geometry
since it also true (but not well-known or appreciated) that,
as you said,
"You cannot apply a Euclidean interpretation of lengths and angles.

!"

PHY-101 students who understand position-vs-time graphs have been able
interpret aspects of the geometry of a Galilean spacetime diagram,
although they are likely unaware of its underlying geometry.

My spacetime diagrammer www.desmos.com/calculator/4765550a89 (by tuning the E-slider)
is an attempt to help connect Euclidean geometry
and the spacetime geometries associated with Galilean relativity and Special Relativity.