Q216 / 270 · NDA 2026 (Phase 1) · Mathematics
Passage: Let $\vec{a}$, $\vec{b}$, $\vec{c}$, $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, $\vec{a}+\vec{b}+\vec{c}$ be unit vectors.
Question: What is the angle between $\vec{a}$ and $\vec{c}$ ?
Question: What is the angle between $\vec{a}$ and $\vec{c}$ ?
- A$\dfrac{\pi}{6}$
- B$\dfrac{\pi}{4}$
- Correct answerC$\dfrac{\pi}{2}$
- D$\dfrac{2\pi}{3}$
Solution
Given $|\vec{a}| = |\vec{b}| = |\vec{c}| = 1$
$|\vec{a}+\vec{b}| = 1 \Rightarrow |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a}\cdot\vec{b} = 1 \Rightarrow 1 + 1 + 2\vec{a}\cdot\vec{b} = 1 \Rightarrow \vec{a}\cdot\vec{b} = -\dfrac{1}{2}$
$|\vec{b}+\vec{c}| = 1 \Rightarrow |\vec{b}|^2 + |\vec{c}|^2 + 2\vec{b}\cdot\vec{c} = 1 \Rightarrow 1 + 1 + 2\vec{b}\cdot\vec{c} = 1 \Rightarrow \vec{b}\cdot\vec{c} = -\dfrac{1}{2}$
$|\vec{a}+\vec{b}+\vec{c}| = 1 \Rightarrow |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) = 1$
$1 + 1 + 1 + 2\left(-\dfrac{1}{2} - \dfrac{1}{2} + \vec{c}\cdot\vec{a}\right) = 1$
$3 + 2(-1 + \vec{c}\cdot\vec{a}) = 1$
$1 + 2\vec{c}\cdot\vec{a} = 1 \Rightarrow \vec{c}\cdot\vec{a} = 0$
Since the dot product of $\vec{a}$ and $\vec{c}$ is $0$, the angle between them is $\dfrac{\pi}{2}$.
Answer: $\dfrac{\pi}{2}$
$|\vec{a}+\vec{b}| = 1 \Rightarrow |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a}\cdot\vec{b} = 1 \Rightarrow 1 + 1 + 2\vec{a}\cdot\vec{b} = 1 \Rightarrow \vec{a}\cdot\vec{b} = -\dfrac{1}{2}$
$|\vec{b}+\vec{c}| = 1 \Rightarrow |\vec{b}|^2 + |\vec{c}|^2 + 2\vec{b}\cdot\vec{c} = 1 \Rightarrow 1 + 1 + 2\vec{b}\cdot\vec{c} = 1 \Rightarrow \vec{b}\cdot\vec{c} = -\dfrac{1}{2}$
$|\vec{a}+\vec{b}+\vec{c}| = 1 \Rightarrow |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) = 1$
$1 + 1 + 1 + 2\left(-\dfrac{1}{2} - \dfrac{1}{2} + \vec{c}\cdot\vec{a}\right) = 1$
$3 + 2(-1 + \vec{c}\cdot\vec{a}) = 1$
$1 + 2\vec{c}\cdot\vec{a} = 1 \Rightarrow \vec{c}\cdot\vec{a} = 0$
Since the dot product of $\vec{a}$ and $\vec{c}$ is $0$, the angle between them is $\dfrac{\pi}{2}$.
Answer: $\dfrac{\pi}{2}$
Hence, the correct answer is option: 'C'.