Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ be such that $\vec{c} - 3\vec{a} = -\vec{b} - \vec{a}$. Then the points with position vectors as $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are:
coplanar
collinear
non-coplanar
None of these
Step-by-Step Solution
Key Concept: Vectors are coplanar if they satisfy a linear dependence relation where scalar coefficients sum to zero.
Given $\vec{a} + \vec{b} + \vec{c} - 3\vec{d} = 0$ and $1 + 1 + 1 - 3 = 0$, we can write $\vec{a} + \vec{b} + \vec{c} = 3\vec{d}$. This means $\vec{a}$, $\vec{b}$, and $\vec{c}$ can be expressed as a linear combination that satisfies the same scalar relation, proving they are coplanar vectors.
Correct Answer: 1