Maths · Components of a vector in two dimensions and three-dimensional spaces
Let be the distinct real numbers. The vectors are
Let \( \alpha, \beta, \gamma \) be the distinct real numbers. The vectors \( \boldsymbol{\alpha} \hat{\boldsymbol{i}}+\boldsymbol{\beta} \hat{\boldsymbol{j}}+\boldsymbol{r} \hat{\boldsymbol{k}} \) \( \beta \hat{i}+r \hat{j}+\alpha \hat{k} ; \gamma \hat{i}+\alpha \hat{j}+\beta \hat{k} \) are
- A. collinear
- B. form an isosceles triangle
- C. right angled triangle
- D. equilatetral triangle
Step-by-step solution
The three vectors are position vectors of points A(α,β,γ), B(β,γ,α), C(γ,α,β). Compute side lengths: AB = √[(β-α)²+(γ-β)²+(α-γ)²], BC = √[(γ-β)²+(α-γ)²+(β-α)²], CA = √[(α-γ)²+(β-α)²+(γ-β)²]. All are equal, so the triangle is equilateral.