What is the mutual position of vectors a and b if: 1^° |a| +…

What is the mutual position of vectors $\vec{a}$ and $\vec{b}$ if: $1^\circ\quad |\vec{a}| + |\vec{b}| = |\vec{a} + \vec{b}| \qquad\qquad 2^\circ\quad |\vec{a}| - |\vec{b}| = |\vec{a} - \vec{b}|$.

Step 1: Case $1^\circ$: $|\vec{a}| + |\vec{b}| = |\vec{a} + \vec{b}|$

Both sides are non-negative, so we may square them: $\left(|\vec{a}| + |\vec{b}|\right)^2 = |\vec{a} + \vec{b}|^2$.

\[ |\vec{a}|^2 + 2|\vec{a}||\vec{b}| + |\vec{b}|^2 = |\vec{a}|^2 + 2\,\vec{a}\cdot\vec{b} + |\vec{b}|^2 \]

Simplifying: \[2|\vec{a}||\vec{b}| = 2\,\vec{a}\cdot\vec{b} \quad\Longrightarrow\quad |\vec{a}||\vec{b}| = \vec{a}\cdot\vec{b}.\]

Since by definition $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$, we get (for non-zero vectors)

\[ |\vec{a}||\vec{b}|\cos\theta = |\vec{a}||\vec{b}| \quad\Longrightarrow\quad \cos\theta = 1 \quad\Longrightarrow\quad \theta = 0. \]

Conclusion: the vectors $\vec{a}$ and $\vec{b}$ are codirectional (parallel, pointing in the same direction). This also follows directly from the triangle inequality: equality $|\vec{a}| + |\vec{b}| = |\vec{a} + \vec{b}|$ holds if and only if the vectors are collinear and point in the same direction.

\[\vec{a} \uparrow\uparrow \vec{b}\]

Step 2: Case $2^\circ$: $|\vec{a}| - |\vec{b}| = |\vec{a} - \vec{b}|$

The right-hand side is non-negative, so the condition can only hold when $|\vec{a}| \geq |\vec{b}|$. Square both sides: $\left(|\vec{a}| - |\vec{b}|\right)^2 = |\vec{a} - \vec{b}|^2$.

\[ |\vec{a}|^2 - 2|\vec{a}||\vec{b}| + |\vec{b}|^2 = |\vec{a}|^2 - 2\,\vec{a}\cdot\vec{b} + |\vec{b}|^2 \]

Simplifying: \[-2|\vec{a}||\vec{b}| = -2\,\vec{a}\cdot\vec{b} \quad\Longrightarrow\quad |\vec{a}||\vec{b}| = \vec{a}\cdot\vec{b}\]

\[ |\vec{a}||\vec{b}|\cos\theta = |\vec{a}||\vec{b}| \quad\Longrightarrow\quad \cos\theta = 1 \quad\Longrightarrow\quad \theta = 0. \]

So again the vectors are codirectional. We must also check consistency with the original equation: for $\vec{a} \uparrow\uparrow \vec{b}$ we have $|\vec{a} - \vec{b}| = \big||\vec{a}| - |\vec{b}|\big|$, so the condition additionally requires

\[ |\vec{a}| - |\vec{b}| \geq 0. \]

Thus $\vec{a}$ and $\vec{b}$ must be codirectional and $|\vec{a}| \geq |\vec{b}|$: $\vec{b}$ points in the same direction as $\vec{a}$ but is not longer.

\[\vec{a} \uparrow\uparrow \vec{b} \quad \text{and} \quad |\vec{a}| \geq |\vec{b}|\]