I'm currently taking a university course in Linear Algebra and Matrix Theory. A recent problem set included a question that asked,
What can you say about two nonzero vectors $\vec{\alpha}$ and $\vec{\beta}$ that satisfy the equation:
$$\|\vec{\alpha}+\vec{\beta}\| \ = \ \|\vec{\alpha}\| + \|\vec{\beta}\| \ $$
$$\vec{\alpha},\vec{\beta} \in \mathbb{R}^n$$
I am attempting to solve this by finding a solution from this equation derived from the law of cosines:
$$\|\vec{\alpha}+\vec{\beta}\|^2 \ = \ \|\vec{\alpha}\|^2 + \|\vec{\beta}\|^2 – \ 2\|\vec{\alpha}\| \|\vec{\beta}\|\cos(\pi-\theta)$$
…so far I have been unable to find a valid solution and am tempted to assert that there exists no $\vec{\alpha}$ and $\vec{\beta}$ for which that equation is true.
Is there any case in which the magnitude of the sum of two vectors equals the sum of the magnitudes?
Best Answer
Working from your point of view you want, for nonzero $\vec\alpha$ and $\vec\beta$, $$ (\|\vec\alpha\|+\|\vec\beta\|)^2=\|\vec\alpha+\vec\beta\|^2=\|\vec\alpha\|^2+\|\vec\beta\|^2-2\|\vec\alpha\|\, \|\vec\beta\|\,\cos(\pi-\theta). $$ So you need $\cos(\pi-\theta)=-1$, which is exactly $\theta=0$. So the two vectors are colinear.