Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §11.1 Vectors and Dot Products, #42 · Multivariable Calculus · Vectors & geometry

The vectors … form a triangle if

The vectors $A, B, C$ form a triangle if $A + B + C = 0$ . The triangle inequality $∣ A + B ∣ \leq ∣ A ∣ + ∣ B ∣$ says that any one side length is less than ______. The proof comes from Schwarz:
$|\mathbf{A}+\mathbf{B}|^2=\mathbf{A}\cdot\mathbf{A}+2\mathbf{A}\cdot\mathbf{B}+\mathbf{B}\cdot\mathbf{B}\le|\mathbf{A}|^2+\text{______}+|\mathbf{B}|^2=(|\mathbf{A}|+|\mathbf{B}|)^2$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §11.1 Vectors and Dot Products, #42

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