Calculate the area of the parallelogram defined by vectors 𝑢⃗ = (2, 1) and 𝑣 = (1, 3).

Question

star233 · Accepted Answer

To calculate the area of the parallelogram defined by vectors

𝐮 = (\begin{matrix} 2 \\ 1 \end{matrix})

and

𝐯 = (\begin{matrix} 1 \\ 3 \end{matrix})

, we can use the cross product.
The cross product of two vectors

𝐮 = (\begin{matrix} u_{1} \\ u_{2} \end{matrix})

and

𝐯 = (\begin{matrix} v_{1} \\ v_{2} \end{matrix})

in two-dimensional space is given by the formula:

𝐮 \times 𝐯 = u_{1} v_{2} - u_{2} v_{1}

.
In this case,

𝐮 = (\begin{matrix} 2 \\ 1 \end{matrix})

and

𝐯 = (\begin{matrix} 1 \\ 3 \end{matrix})

. Let's calculate the cross product:

𝐮 \times 𝐯 = (2) (3) - (1) (1) = 6 - 1 = 5

.
The magnitude of the cross product represents the area of the parallelogram formed by the two vectors. Therefore, the area of the parallelogram is 5 square units.
Hence, the area of the parallelogram defined by vectors

𝐮 = (\begin{matrix} 2 \\ 1 \end{matrix})

and

𝐯 = (\begin{matrix} 1 \\ 3 \end{matrix})

is 5 square units.

Calculate the area of the parallelogram defined by

Answered question

Answer & Explanation

New Questions in Calculus 1