Vector Algebra
Scalar Triple Product – Numerical
Grade 12
Question:
<p>Let \(\vec{a}=2\hat{i}+3\hat{j}-\hat{k}\), \(\vec{b}=\hat{i}-\hat{j}+2\hat{k}\),
\(\vec{c}=2\hat{i}+\hat{j}-\hat{k}\). If
\([\vec{a}+\vec{b},\,\vec{b}+\vec{c},\,\vec{c}+\vec{a}]=\lambda[\vec{a},\vec{b},\vec{c}]\),
find \(\lambda\).</p>
Step-by-Step Solution
Key Concept: The identity [a+b, b+c, c+a] = 2[a,b,c] holds for any three vectors a,b,c.
This is a standard identity:
$[\vec{a}+\vec{b},\;\vec{b}+\vec{c},\;\vec{c}+\vec{a}]=2[\vec{a},\vec{b},\vec{c}]$
Proof sketch: expand the triple product — each of the 8 terms either contributes
$[\vec{a},\vec{b},\vec{c}]$ (twice) or vanishes (repeated vector $\Rightarrow$ zero).
Therefore $\lambda=\boxed{2}$.
Correct Answer: 2