Question

The adjacent side of parallelogram is represented by vectors $2\hat i + 3\hat j$ and $\hat i + 4\hat j$. The area of the parallelogram is
(A) $5$ units
(B) $3$ units
(C) $8$ units
(D) $11$ units

Answer 1

Hint
The area of parallelogram and the area of parallelogram vectors both are different. And both are having different area formulas. Here, the given data is in vector form, so by creating the matrix form for the given vector and then by combining the two vectors into one vector form and by using this one vector, the area is determined.
The area of the parallelogram vectors is given by,
$\Rightarrow A = \sqrt {{a^2} + {b^2} + {c^2}} $
Where, $A$ is the area of the parallelogram vectors, $a$ is the coefficient of $\hat i$, $b$ is the coefficient of $\hat j$ and $c$ is the coefficient of $\hat k$.

Complete step by step answer
Given that,
The one side of the vector equation is, $a = 2\hat i + 3\hat j + 0\hat k$
The other side of the vector equation is, $b = \hat i + 4\hat j + 0\hat k$
Now, by creating the given vector in the matrix form by,
Now performing the matrix multiplication, then
$\Rightarrow A = \hat i\left[ {\left( {0 \times 3} \right) - \left( {4 \times 0} \right)} \right] - \hat j\left[ {\left( {0 \times 2} \right) - \left( {1 \times 0} \right)} \right] + \hat k\left[ {\left( {2 \times 4} \right) - \left( {1 \times 3} \right)} \right]$
On multiplying the terms in the above equation, then the above equation is written as,
$\Rightarrow A = \hat i\left[ {0 - 0} \right] - \hat j\left[ {0 - 0} \right] + \hat k\left[ {8 - 3} \right]$
Now, by adding the terms inside the bracket, then the above equation is written as,
$\Rightarrow A = 0\hat i - 0\hat j + 5\hat k$
From this equation, the values of $a$, $b$ and $c$ are $0$, $0$ and $5$ respectively.
Now, The area of the parallelogram vectors is given by,
$\Rightarrow A = \sqrt {{a^2} + {b^2} + {c^2}} $
By substituting the values of $a$, $b$ and $c$ in the above equation, then the above equation is written as,
$\Rightarrow A = \sqrt {{0^2} + {0^2} + {5^2}} $
By squaring the terms inside the root, then the above equation is written as,
$\Rightarrow A = \sqrt {0 + 0 + 25} $
By adding the terms, then the above equation is written as,
$\Rightarrow A = \sqrt {25} $
By using the square root, then the above equation is written as,
$\Rightarrow A = 5$ units.
Hence, the option (A) is the correct answer.

Note
To solve this problem the condition of the matrix multiplication in one matrix must be known. There are several conditions that must be followed during the multiplications. That matrix multiplication is used to combine the two or more-vector equations to the single vector equation.