Sum of vector magnitudes

Question

I am trying to solve: Let, $$\vec a, \vec b ∈ \mathbb{R}^2 $$ $$\vec a = (a_1, a_2), \vec b = (b_1, b_2)$$ $$\vec c = \vec a + \vec b $$ Find vectors a, b, so that: $$ \vert \vec c \vert = \vert \vec a \vert + \vert \vec b \vert $$ $$ \sqrt {(a_1 + b_1)^2 + (a_2+b_2)^2} = \sqrt {a_1^2+a_2^2} + \sqrt {b_1^2+b_2^2} \text{ (1)}$$ Solving it, I get $$ (a_2*b_1 - a_1*b_2)^2 = 0$$ $$ a_2*b_1 = a_1*b_2 $$

$$\text{When I pick } a = [7, 1], b=[-1, x],$$ $$ \text{I get } a = [7, 1], b = [-1, -\frac{1}{7}]$$ Substituting these values back in (1) does not give an equality. I can see that b is a vector in the opposite direction of a, so it sort of makes sense intuitively why the sum of a and b doesn't have the same magnitude as |a| + |b|. However, since solving the equation gives me the value for b2, I am confused about the result and why aren't both sides of (1) equal. Could someone please offer an explanation?

Answer 1

hint

For the vectors:

$$|a+b| \leq |a|+|b|$$

When happen equality?

See this: Proof for triangle inequality for vectors

The equality happens when $a=k\cdot b$ with $k>0$.

Ps.: You are missing some evaluations. You are saying that the vectors $a$ and $b$ are in opposite directions what is not true (see above: $k>0$).