Question

In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$. $P_1=(-2,0,2), \quad P_2=(2,1,-1), \quad P_3=(2,-1,2)$

   In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$.
 $P_1=(-2,0,2), \quad P_2=(2,1,-1), \quad P_3=(2,-1,2)$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 8, Problem 48

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Step 1

To find $\overrightarrow{P_1P_2}$, subtract the coordinates of $P_1$ from $P_2$: \[ \overrightarrow{P_1P_2} = P_2 - P_1 = (2, 1, -1) - (-2, 0, 2) = (2 + 2, 1 - 0, -1 - 2) = (4, 1, -3). \] Similarly, to find $\overrightarrow{P_1P_3}$, subtract the coordinates of Show more…

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In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$. $P_1=(-2,0,2), \quad P_2=(2,1,-1), \quad P_3=(2,-1,2)$

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Key Concepts

Area of a Parallelogram Using the Cross Product

The area of a parallelogram formed by two adjacent vectors can be found by computing the magnitude of their cross product. This method uses the geometric interpretation of the cross product to directly relate the algebraic operations on vectors to the spatial area they span.

Vector Subtraction

Vector subtraction is a fundamental operation in vector geometry where one vector is subtracted from another to produce a third vector representing the relative displacement between their endpoints. This operation is essential for defining vectors that represent sides or edges in geometric figures.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a new vector perpendicular to both original vectors. The magnitude of this product reflects the area of the parallelogram spanned by the two vectors, linking vector algebra with geometric properties.

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