Vector Algebra
Linear Dependence of Vectors
Grade 12
Question:
<p>If \(\vec{a},\vec{b},\vec{c}\) are three non-zero vectors satisfying the condition \(\vec{a}+\vec{b}+\vec{c}=\vec{0}\), then which of the following hold(s)?</p>
\(\vec{a}\times\vec{b}=\vec{b}\times\vec{c}=\vec{c}\times\vec{a}\)
\(\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{c}=\vec{c}\cdot\vec{a}\) (not necessarily equal)
\(|\vec{a}|^2=\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}\) type identity holds
\([\vec{a},\vec{b},\vec{c}]=0\)
Step-by-Step Solution
Key Concept: From a+b+c=0: cross with a gives a \times b=a \times (-b-c)... and scalar triple product is always 0 when three vectors sum to zero (they're coplanar).
From $\vec{a}+\vec{b}+\vec{c}=\vec{0}\Rightarrow\vec{c}=-\vec{a}-\vec{b}$.
Option A: $\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}$: Cross $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ with $\vec{a}$: $\vec{a}\times\vec{a}+\vec{b}\times\vec{a}+\vec{c}\times\vec{a}=\vec{0}\Rightarrow\vec{b}\times\vec{a}+\vec{c}\times\vec{a}=\vec{0}\Rightarrow\vec{a}\times\vec{b}=\vec{c}\times\vec{a}$. Similarly $\vec{a}\times\vec{b}=\vec{b}\times\vec{c}$. ✓ (A)
Option D: $[\vec{a},\vec{b},\vec{c}]=(\vec{a}\times\vec{b})\cdot\vec{c}$. Since $\vec{c}=-\vec{a}-\vec{b}$: $=(\vec{a}\times\vec{b})\cdot(-\vec{a}-\vec{b})=0$. ✓ (C matches)
Answer: AC
Correct Answer: AC