Question

If the vectors $\left( {\hat i + \hat j + \hat k} \right)$ and $3\hat i$forms two sides of a triangle, then area of the triangle is
(1) \sqrt 3 $$$$unit
(2) 2\sqrt 3 $$$$unit
(3) \dfrac{3}{{\sqrt 2 }}$$$$unit
(4) 3\sqrt 2 $$$$unit

Answer 1

If two sides of a triangle are given in vector form, then the area of the triangle is given by half time of the magnitude of the cross product of two sides. If $\overrightarrow a$ and $\overrightarrow b$ are the vectors representing the side of a triangle then its area is given by$\dfrac{1}{2}\left| {\overrightarrow a \times \overrightarrow b } \right|$.
Cross product: Multiplication of vectors can be given in two ways. First way is a scalar or dot product. Second one is known as a vector product or cross product.
Cross product of two vectors gives the area vector of a parallelogram formed by the two vectors. Then the area of the triangle is half of the area vector of a parallelogram.

Complete step by step answer:
Given: two sides of a triangle. Let PQR be the triangle. Then two sides of triangle is given by
$\overrightarrow {PQ} = \left( {\hat i + \hat j + \hat k} \right)$ and $\overrightarrow {QR} = 3\hat i$
Area of triangle in vector form is given by
= \dfrac{1}{2}$$$$\left( {magnitude{\text{ }}cross{\text{ }}product{\text{ }}of{\text{ }}two{\text{ }}vectors} \right)
=$\dfrac{1}{2}\left| {\overrightarrow {PQ} \times \overrightarrow {QR} } \right|$
=$\dfrac{1}{2}\left| {\left( {\hat i + \hat j + \hat k} \right) \times 3\hat i} \right|$
=$\dfrac{1}{2}\left| {\left( {\hat i + \hat j + \hat k} \right) \times \left( {3\hat i + 0\hat j + 0\hat k} \right)} \right|$
$\overrightarrow {PQ} \times \overrightarrow {QR}$=$\left( {\hat i + \hat j + \hat k} \right) \times 3\hat i$
= \left| {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\\ 1&1&1 \\\ 3&0&0 \end{array}} \right|
=$\hat i(0 - 0) - \hat j(0 - 3) + \hat k(0 - 3)$
=$3\hat j - 3\hat k$
Hence, area of triangle PQR= $\dfrac{1}{2}\left| {3\hat j - 3\hat k} \right|$
=$\dfrac{1}{2}\left( {\sqrt {{3^2} + {{( - 3)}^2}} } \right)$
=$\dfrac{{3\sqrt 2 }}{2}$
=\dfrac{3}{{\sqrt 2 }}$$$$unit

So, the correct answer is “Option C”.

Note:
For cross product, vectors should be written in the form of$\left( {x\hat i + y\hat j + z\hat k} \right)$. if any two coordinates are given then consider the third coordinate as zero. Because for a cross product of two vectors all three coordinates $\left( {x,y,z} \right)$are necessary. Otherwise we don’t get the correct answer. Students make mistakes in writing a vector in the form of $\left( {x\hat i + y\hat j + z\hat k} \right)$. In question the second vector has only one coordinate. so we should write this vector in the form of$\left( {x\hat i + y\hat j + z\hat k} \right)$. Remember one thing that cross product of two vectors can only defined in three dimensional coordinate plane