Find the area of the parallelogram with vertices A(-3, 0), B(-1 , 3), C(5, 2), and D(3, -1).

Question

Janet Young · Accepted Answer

From the given points, we may choose any three of them such and find a two vector that express the sides of the parallelogram "or one side and a diagonal of the parallelogram", we may choose for example the points B,C and D
Moreover, we can choose point B to be the common point for the two sides "the initial point of the two vector", thus we need to find vector BD and vector BC, using equation as follows

B C = < 5 + 1, 2 - 3 >

= < 6, - 1 >

And, the other vector side BD is

B D = < 3 + 1, - 1 - 3 >

= < 4, - 4 >

And the cross product of both vectors is

B C \times B D = < 6, - 1 > \times < 4, - 4 >

= [\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & - 1 & 0 \\ 4 & - 4 & 0 \end{matrix}]

= \hat{i} [\begin{matrix} - 1 & 0 \\ 4 & 0 \end{matrix}] - \hat{j} [\begin{matrix} 6 & 0 \\ 4 & 0 \end{matrix}] + \hat{k} [\begin{matrix} 6 & - 1 \\ 4 & - 4 \end{matrix}]

= \hat{i} - 0 \hat{j} + \hat{k} ((6) (- 4) - (1) (- 4))

= - 20 \hat{k}

Knowing, the cross product of the two vectors of the parallelogram we can use equation to find the area

A r e a = | - 20 \hat{k} |

= 20

Jenny Bolton · Accepted Answer

Assume that the parallelogram plan is the xy plan
The four vertices are $A (- 3, 0), B (- 1, 3), C (5, 2)$ and $D (3, - 1)$
We can get the are by $| \overset{―}{A B} \times \overset{―}{B C} |$
such that $| \overset{―}{A B} |$ is the vector starting at the point A and ending with B and $\overset{―}{B C}$ is the vector starting at the point B and ending with C
$| \vec{A B} | = (- 1 - (- 3)) i + (3 - 0) j = 2 i + 3 j$
$| \vec{B C} | = (5 - (- 1)) i + (2 - 3) j = 6 i - j$
regarding that ${i, j, k}$ are perpendicular unit vector on the ${x, y, z}$
So
$| \overset{―}{A B} \times | \overset{―}{B C} | = | (2 i + 3 j) \times (6 i - j) ∣$
$| \overset{―}{A B} \times | \overset{―}{B C} | = | (0 - 2 k - 18 k + 0) |$
$| \overset{―}{A B} \times | \overset{―}{B C} | = | - 20 k ∣ = 20$
hence the area is 20 units

Find the area of the parallelogram with vertices A(-3, 0),

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