Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )$
A)15
B) $0$
C) $\sqrt { 3 }$
D)21
E) $5 \sqrt { 35 }$

Question

Find the area of the parallelogram that has the vectors as adjacent sides.  $\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )$ 
A)15
B) $0$
C) $\sqrt { 3 }$
D)21
E) $5 \sqrt { 35 }$

Quizplus · Accepted Answer

The area of a parallelogram formed by vectors $\mathbf{u}$ and $\mathbf{v}$ is given by the magnitude of their cross product, $\|\mathbf{u} \times \mathbf{v}\|$. The cross product $\mathbf{u} \times \mathbf{v}$ is calculated as follows:$$\mathbf{u} \times \mathbf{v} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \-1 & 4 & -3 \-3 & 3 & 4 \\end{vmatrix}= (4\cdot4 - (-3)\cdot3)\mathbf{i} - ((-1)\cdot4 - (-3)\cdot(-3))\mathbf{j} + ((-1)\cdot3 - 4\cdot(-3))\mathbf{k}= (16 + 9)\mathbf{i} - (-4 - 9)\mathbf{j} + (-3 + 12)\mathbf{k}= 25\mathbf{i} + 13\mathbf{j} + 9\mathbf{k}$$The magnitude of this vector is:$$\|\mathbf{u} \times \mathbf{v}\| = \sqrt{25^2 + 13^2 + 9^2} = \sqrt{625 + 169 + 81} = \sqrt{875} = 5\sqrt{35}$$Therefore, the area of the parallelogram is $5\sqrt{35}$, which corresponds to choice E.

Find the Area of the Parallelogram That Has the Vectors u=(−1,4,−3),v=(−3,3,4)\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )u=(−1,4,−3),v=(−3,3,4)

Find the Area of the Parallelogram That Has the Vectors $\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )$