Vector Algebra
Scalar Triple Product
Grade None
Question:
<p>Let \(\vec{a},\vec{b}\) be two non-zero non-collinear vectors. Then for any scalar \(\lambda\neq0\), \(\vec{a}=\lambda\vec{b}\) if and only if</p>
<li>\(\vec{a}\times\vec{b}=\vec{0}\)</li>
<li>\(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\)</li>
<li>\(\vec{a}\times\vec{b}=\vec{0}\) or \(\vec{a}\cdot\vec{b}=0\)</li>
<li>\(\exists\lambda\in\mathbb{R}:\vec{a}=\lambda\vec{b}\)</li>
Step-by-Step Solution
Key Concept: Two vectors are parallel iff their cross product is zero, iff one is a scalar multiple of the other.
Parallel (collinear) vectors satisfy $\vec{a}\times\vec{b}=\vec{0}$ (since $\sin\theta=0$) and equivalently $\vec{a}=\lambda\vec{b}$ for some scalar $\lambda$.
Option A: $\vec{a}\times\vec{b}=\vec{0}\Leftrightarrow\vec{a}\parallel\vec{b}$. ✓
Option D: $\vec{a}=\lambda\vec{b}\Leftrightarrow\vec{a}\parallel\vec{b}$. ✓
Answer: AD
Correct Answer: AD