Vectors
Vectors
Allen Star Batch
Grade 12
Question:
The position vectors of the vertices $A, B$ and $C$ of a triangle are three unit vectors $\vec{a}, \vec{b}$ and $\vec{c}$. A vector $\vec{d}$ is such that $\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c}$ and $\vec{d} = \lambda(\vec{b} + \vec{c})$, then triangle $ABC$ is:
Acute angled
Obtuse angled
Right angled
None of these
Step-by-Step Solution
Key Concept: Equal angles with three position vectors imply a perpendicularity condition leading to a right angle.
From $\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c}$, suppose $\lambda(\vec{b} \cdot \vec{c}) = \lambda(\vec{b} \cdot \vec{c})$. Then $1 + \vec{b} \cdot \vec{c} = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$ leads to $1 - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} - \vec{a} \cdot \vec{c} = 0$. This rearranges to $(\vec{a} - \vec{c}) \cdot (\vec{a} - \vec{b}) = 0$, showing $(\vec{a} - \vec{c})$ is perpendicular to $(\vec{a} - \vec{b})$. Therefore the triangle is right-angled.
Correct Answer: 3