NIMCET 2013 — Mathematics PYQ

Step 1: Define the Side Vectors

In a triangle $ABC$, if the position vectors of vertices $A,B,$ and $C$ are $\vec{a},\vec{b},$ and $\vec{c}$ respectively, we can define two adjacent sides as vectors:

• Side $\overrightarrow{AB}=\vec{b}-\vec{a}$

• Side $\overrightarrow{AC}=\vec{c}-\vec{a}$

Step 2: Apply the Area Formula

The area of a triangle with adjacent sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is given by:

Step 3: Expand the Cross Product

Substitute the position vectors into the formula:

Now, distribute the cross product:

Step 4: Simplify the Expression

Using the properties of cross products:

• $\vec{a}\times \vec{a}=0$ (Cross product of a vector with itself is zero)

• $-\vec{b}\times \vec{a}=\vec{a}\times \vec{b}$ (Anticommutative property)

• $-\vec{a}\times \vec{c}=\vec{c}\times \vec{a}$

Substitute these back into the expression:

Rearranging the terms to match the standard form:

Final Answer: The area is $\frac{1}{2}|\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}|$.