Question:
<p>Let \(\vec{a},\vec{b},\vec{c}\) be non-zero vectors with no two parallel.
If \((\vec{a}+2\vec{b})\perp\vec{c}\) and \((\vec{a}+3\vec{b})\perp\vec{c}\),
which must be true?</p>
\(\vec{a}\perp\vec{c}\)
\(\vec{b}\perp\vec{c}\)
\(\vec{a}\perp\vec{b}\)
\(\vec{a}\perp\vec{c}\) and \(\vec{b}\perp\vec{c}\)
Step-by-Step Solution
Key Concept: Subtract the two perpendicularity conditions to isolate b \cdot c = 0. Then substitute back to get a \cdot c = 0.
$(\vec{a}+2\vec{b})\cdot\vec{c}=0\Rightarrow\vec{a}\cdot\vec{c}+2\vec{b}\cdot\vec{c}=0$...(1)
$(\vec{a}+3\vec{b})\cdot\vec{c}=0\Rightarrow\vec{a}\cdot\vec{c}+3\vec{b}\cdot\vec{c}=0$...(2)
(2)-(1): $\vec{b}\cdot\vec{c}=0\Rightarrow\vec{b}\perp\vec{c}$.
Sub in (1): $\vec{a}\cdot\vec{c}=0\Rightarrow\vec{a}\perp\vec{c}$.
Answer: D (both a⊥c and b⊥c).
Correct Answer: D