Vectors and coordinate geometry

Find the length of the vector $v = (2, -3, 6)$.

Compute $(1, 2, 3) \cdot (4, -5, 6)$.

For which number $k$ is $(2, k)$ perpendicular to $(3, -6)$?

Find the angle, in degrees, between $u = (1, 1, 0)$ and $v = (1, 0, 1)$.

Let $u, v, w$ be vectors in $\R^3$ and $t$ a number. Which of these expressions are vectors? (Choose all that apply.)

For non-zero vectors $u$ and $v$, when is $|u + v| = |u| + |v|$?

At which value of $t$ does the line $(1, 0, 0) + t\,(1, 2, 2)$ meet the plane $x + y + z = 6$?

In the vector proof that the diagonals of a parallelogram $ABCD$ bisect each other, which equation expresses '$ABCD$ is a parallelogram' (with vertices in this order)?

In Mathlib, ⟪u, v⟫ is the dot product and ‖u‖ the length. Which statement is the Cauchy–Schwarz inequality?

Write projection(u, v) that returns the projection of $u$ onto a non-zero vector $v$, $\dfrac{u\cdot v}{v \cdot v}\,v$, as a list of Fractions (vectors are lists of integers of the same length, in any dimension). Raise ValueError if $v$ is the zero vector.