Question

If $a = 2i - 3j + k,\;{\text{b = - i + k, c = 2j - k,}}$ find the area of the parallelogram having diagonals $a + b$ and $b + c$ A.Area $= \dfrac{1}{4}\sqrt {21}$ B.Area $= \dfrac{1}{4}\sqrt {19}$ C.Area $= \dfrac{1}{2}\sqrt {19}$ D.Area $= \dfrac{1}{2}\sqrt {21}$

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Hint: First of all we will assume two diagonals then will find the diagonals then will place the value of diagonals in the area formula. Will use a cross-product method to find the diagonal product and then will find magnitude.

Let us take the given vertices –
$a = 2i - 3j + k,\;{\text{b = - i + k, c = 2j - k,}}$
Let us assume that diagonals ${d_1} = a + b$ and ${d_2} = b + c$
Now, given that the diagonals of the parallelogram are –
Find $a + b$
Place the values in the above expression –
$a + b = (2i - 3j + k) + ( - i + k)$
Open the brackets and simplify. Remember when there is a positive sign outside the bracket then there is no change in the signs of the terms inside the bracket when you open it.
$a + b = 2i - 3j + k - i + k$
Make the pair of like terms –
$\Rightarrow a + b = \underline {2i - i} - 3j\underline { + k + k}$
Simplify the above equation –
$\Rightarrow a + b = i - 3j + 2k$
$\Rightarrow {d_2} = i - 3j + 2k$ .... (A)
Similarly for second diagonal
$b + c = - i + k + 2j - k$
Make the pair of like terms –
$b + c = - i + 2j\underline { - k + k}$
Terms with the same value and opposite sign cancel each other. Simplify the above equation –
$b + c = - i + 2j$
Therefore, ${d_2} = - i + 2j$ .... (B)
Now, the area of the parallelogram is
$A = \dfrac{1}{2}\left| {{d_1} \times {d_2}} \right|$
Place values form equation (A) and (B)
$A = \dfrac{1}{2}\left| {(i - 3j + 2k) \times ( - i + 2j)} \right|$ ..... (C)
Now find cross product –
$\left| {\begin{array}{*{20}{c}} i&j&k \\ 1&{ - 3}&2 \\ { - 1}&2&0 \end{array}} \right|$
Open the determinant –
$= i( - 4) - j(2) + k(2 - 3)$
Simplify the above equation – when you subtract a bigger number from smaller there will be a negative sign in resultant value.
 $= 4i - 2j - k$
Now find the mode of the above vector expression –
$\left| {4i - 2j - k} \right| = \sqrt {{4^2} + {{( - 2)}^2} + {{( - 1)}^2}}$
Remember the square of the negative number also gives the positive value. Since minus multiplied with minus gives plus.
$\left| {4i - 2j - k} \right| = \sqrt {16 + 4 + 1}$
Simplify the above equation –
$\left| {4i - 2j - k} \right| = \sqrt {21}$ .... (D)
Place above value in equation (C)
$\Rightarrow A = \dfrac{{\sqrt {21} }}{2}$ Sq.units
So, the correct answer is “Option D”.

Note: Parallelogram law:
If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal of parallelogram through the common point represents the sum of the two vectors in both magnitude and direction.