Vector Algebra
Unit Vectors — Dot Product Sum from Magnitude Condition
nta_pyq_2026_jan
Grade None
Question:
For three unit vectors $\vec{a},\vec{b},\vec{c}$ satisfying $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2=9$ and $|2\vec{a}+k\vec{b}+k\vec{c}|=3$, the positive value of $k$ is:
Step-by-Step Solution
Key Concept: Expanding: $6-2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a})=9\Rightarrow\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}=-\frac{3}{2}$. For the symmetric case $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{c}=\vec{c}\cdot\vec{a}=-\frac{1}{2}$.
Positive value of $k=5$.
Correct Answer: 3