Vector Algebra
Summation of Unit Vectors – Numerical
Grade 12
Question:
<p>Let \(\hat{a}_1,\hat{a}_2,\ldots,\hat{a}_n\) be unit vectors in the plane
such that \(\displaystyle\sum_{i=1}^{n}\hat{a}_i=\vec{0}\). Let
\(k=\displaystyle\sum_{i<j}\hat{a}_i\cdot\hat{a}_j\). Find \(2k+n\).</p>
Step-by-Step Solution
Key Concept: Expand |\Sigmaaᵢ|^2 = 0: it equals n + 2\Sigmaᵢ<ⱼ aᵢ \cdot aⱼ = n + 2k = 0.
$\left|\sum_{i=1}^n\hat{a}_i\right|^2=\sum_i|\hat{a}_i|^2+2\sum_{i<j}\hat{a}_i\cdot\hat{a}_j=n+2k=0$.
Therefore $2k+n=\boxed{0}$.
Correct Answer: 0