How to solve the coordinates containing points and vectors in the equation?

Question

Reduce[{
  a == {1, 1}, b == {1, -1},
  (a + λ b) . (a + μ b) == 0, 
  t == λ + μ, n == λ μ
 }, {n, t}
]

(* False *)

Why can't we find a solution?

Answer 1

Try this instead:

eq = With[{a = {1, 1}, b = {1, -1}},
  {(a + λ b) . (a + μ b) == 0, 
  t == λ + μ, n == λ μ
 }] // Simplify

(* Out: {1 + λ μ == 0, t == λ + μ, n == λ μ} *)


Reduce[eq, {n, t}]

(* Out: μ != 0 && λ == - 1 / μ && n == -1 && t == λ + μ *)

Answer 2

• Since there are . in the equation, we need to tell Mathematica some Assumptions to expand the tensor.

$Assumptions = {a ∈ Vectors[2, Reals], 
   b ∈ Vectors[2, Reals], λ ∈ 
    Reals, μ ∈ Reals};
TensorExpand[(a + λ b) . (a + μ b) == 0]

a . a + λ a . b + μ a . b + λ μ b . b == 0.

• So add the domains solved my doubt about this problem.

Reduce[{a == {1, 1}, 
  b == {1, -1}, (a + λ b) . (a + μ b) == 0, 
  t == λ + μ, n == λ μ}, {n ∈ Reals, 
  t ∈ Reals, λ ∈ Reals, μ ∈ 
   Reals, a ∈ Vectors[2, Reals], 
  b ∈ Vectors[2, Reals]}]