Find the area of the parallelogram formed by the points $A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )$, $C ( 3 , - 2,11 )$, and $D ( 8 , - 1,13 )$.
A) 816
B) $2 \sqrt { 5 }$
C) 20
D) $4 \sqrt { 51 }$
E) 18

Question

Quizplus · Accepted Answer

The area of a parallelogram formed by vectors $\vec{AB}$ and $\vec{AC}$ can be found using the cross product of these vectors, $\vec{AB} \times \vec{AC}$, and then finding the magnitude of this cross product. The vectors $\vec{AB}$ and $\vec{AC}$ are found by subtracting the coordinates of points A from B and C, respectively. $\vec{AB} = B - A = (7 - 2, -2 + 3, 7 - 5) = (5, 1, 2)$$\vec{AC} = C - A = (3 - 2, -2 + 3, 11 - 5) = (1, 1, 6)$The cross product $\vec{AB} \times \vec{AC}$ is given by the determinant of the matrix:$$\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \5 & 1 & 2 \1 & 1 & 6 \\end{vmatrix}= \mathbf{i}(1*6 - 2*1) - \mathbf{j}(5*6 - 2*1) + \mathbf{k}(5*1 - 1*1)= 4\mathbf{i} - 28\mathbf{j} + 4\mathbf{k}$$The magnitude of this vector is:$$\sqrt{4^2 + (-28)^2 + 4^2} = \sqrt{16 + 784 + 16} = \sqrt{816} = 4\sqrt{51}$$Therefore, the area of the parallelogram is $4\sqrt{51}$.

Find the Area of the Parallelogram Formed by the Points A(2,−3,5),B(7,−2,7)A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )A(2,−3,5),B(7,−2,7)

Find the Area of the Parallelogram Formed by the Points $A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )$