Understanding Triangle Area Using Position Vectors

This question asks for the formula to calculate the area of a triangle when the position vectors of its vertices are given. Let the vertices of the triangle be A, B, and C, with corresponding position vectors \(\rm \vec{a}\), \(\rm \vec{b}\), and \(\rm \vec{c}\).

Deriving the Area Formula

We can find the area of the triangle by considering two vectors representing two sides of the triangle originating from the same vertex. Let's choose vertex A.

• The vector representing the side AB is given by the difference between the position vectors of B and A: \(\rm \vec{AB} = \vec{b} - \vec{a}\)
• Similarly, the vector representing the side AC is: \(\rm \vec{AC} = \vec{c} - \vec{a}\)

Vector Cross Product and Triangle Area

The magnitude of the cross product of two vectors gives the area of the parallelogram formed by these vectors. The area of the triangle formed by these vectors is half the area of the parallelogram.

Therefore, the area of triangle ABC is:

Area = \(\rm \dfrac{1}{2} |\vec{AB} \times \vec{AC}|\)

Substituting Position Vectors

Now, substitute the expressions for \(\rm \vec{AB}\) and \(\rm \vec{AC}\) into the area formula:

Area = \(\rm \dfrac{1}{2} |(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})|\)

Expanding the Cross Product

Let's expand the cross product term using the distributive property:

\(\rm (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) = (\vec{b} \times \vec{c}) - (\vec{b} \times \vec{a}) - (\vec{a} \times \vec{c}) + (\vec{a} \times \vec{a})\)

Applying Vector Properties

We use the following properties of the cross product:

• The cross product of a vector with itself is the zero vector: \(\rm \vec{a} \times \vec{a} = \vec{0}\).
• The cross product is anti-commutative: \(\rm \vec{u} \times \vec{v} = -(\vec{v} \times \vec{u})\). This means \(\rm \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\) and \(\rm \vec{a} \times \vec{c} = -(\vec{c} \times \vec{a})\).

Simplifying the Expression

Substitute these properties back into the expanded expression:

\(\rm (\vec{b} \times \vec{c}) - (\vec{b} \times \vec{a}) - (\vec{a} \times \vec{c}) + (\vec{a} \times \vec{a})\)

= \(\rm (\vec{b} \times \vec{c}) - (-(\vec{a} \times \vec{b})) - (-(\vec{c} \times \vec{a})) + \vec{0}\)

= \(\rm \vec{b} \times \vec{c} + \vec{a} \times \vec{b} + \vec{c} \times \vec{a}\)

Final Area Formula

Substituting this simplified expression back into the area formula, we get:

Area = \(\rm \dfrac{1}{2} |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|\)

Comparing with Options

This derived formula matches Option 2 provided in the question.

• Option 1: \(\rm \dfrac{1}{2} (\vec{a}\times \vec{b})\vec{c}\) is dimensionally incorrect and does not represent the area.
• Option 3: \(\rm \vec{a}\times \vec{b} + \vec{b} \times \vec{c} + \vec{c}\times \vec{a}\) is a vector quantity, whereas area is a scalar.
• Option 4: None of these is incorrect as Option 2 is correct.

Thus, the correct representation for the area of the triangle with position vectors \(\rm \vec{a}\), \(\rm \vec{b}\), and \(\rm \vec{c}\) is \(\rm \dfrac{1}{2} |\vec{a}\times \vec{b} + \vec{b} \times \vec{c}+\vec{c}\times \vec{a}|\).