Let $u$ and $v$ be two vectors. Then $|u-v|=|| u|-| v||$ if and only if
Let $u$ and $v$ be two vectors. Then $|u-v|=|| u|-| v||$ if and only if
- $|u|=|v|$
- $u$ and $v$ have the same direction
- $u$ and $v$ have the opposite direction
- $u=v$
Solution
Since, $|u-v|^2=|u|^2+|v|^2-2 \vec{u} \cdot \vec{v}$
and $\| u|-| v||^2=|u|^2+|v|^2-2|\vec{u}||\vec{v}|$
Now, $|u-v|^2=\left.|| u|-| v\right|^2$ only, when
It only happens when $u$ and $v$ have same direction.
$
\begin{aligned}
\vec{u} \cdot \vec{v} & =|\vec{u}||\vec{v}| \\
|\vec{u}| \| \vec{v} \mid \cos \theta & =|\vec{u}||\vec{v}| \Rightarrow \cos \theta=1 \\
\theta & =0
\end{aligned}
$
Asked in: AP EAMCET 2021 (23 Aug Shift 1)
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