Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places. $\mathbf { u } =(-7,9,5), \mathbf { v } = ( - 2 , - 2,5 )$
A) $\sqrt { 9,348 }$ or $96.69$
B) $\sqrt { 4,676 }$ or $68.38$
C) $\sqrt { 4674 }$ or $68.37$
D) $\sqrt { 9,701 }$ or $98.49$
E) $\sqrt { 9,349 }$ or $96.69$

Question

Quizplus · Accepted Answer

The area of a parallelogram formed by two vectors $\mathbf{u}$ and $\mathbf{v}$ is given by the magnitude of the cross product $\|\mathbf{u}	imes\mathbf{v}\|$. Thus, we need to find the cross product of $\mathbf{u}$ and $\mathbf{v}$:
$$\mathbf{u} 	imes \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -7 & 9 & 5 \ -2 & -2 & 5 \end{vmatrix} = (45, 29, 52)$$
Then, the area of the parallelogram is $\|\mathbf{u} 	imes \mathbf{v}\|=\sqrt{45^2+29^2+52^2}=\sqrt{4674}\approx \boxed{68.37}$. Therefore, the best choice is $\boxed{	extbf{(C)}}$.

Find the Area of a Parallelogram That Has the Given u=(−7,9,5),v=(−2,−2,5)\mathbf { u } =(-7,9,5), \mathbf { v } = ( - 2 , - 2,5 )u=(−7,9,5),v=(−2,−2,5)

Find the Area of a Parallelogram That Has the Given $\mathbf { u } =(-7,9,5), \mathbf { v } = ( - 2 , - 2,5 )$