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Enrique C.

Precalculus

4 days, 9 hours ago

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Area of a triangle For the given points $A, B$, and $C$, find the area of the triangle with vertices $\boldsymbol{A}, \boldsymbol{B},$ and $\boldsymbol{C}$ $$A(-1,-5,-3), B(-3,-2,-1), C(0,-5,-1)$$

Chapter 12

Vectors and Vector-Valued Functions

Section 4

Cross Products

are there today. We're going to calculate the area of the triangle defined by a, which is minus one minus five minus three B, which is minus three minus two minus one on DSI, which is zero minus five minus one three points. Define a triangle in a plane, which we can just call a B and C and can collect the area we require to find vectors between them. Andi it's wise to choose vectors. That's infi calculations the most. So if we look at the vector a C, this is C minus a. This chosen because they both have the same second component. So it's zero minus five minus one minus minus one minus five minus three. So they zero plus one, which is one minus five plus five. Which is there a minus one plus three, which is to which we could call you. And then we need to choose our next vector, uh, which we might choose to be a B. But his B minus a just minus three minus two minus one or despise one minus five minus three. His minus three plus one, which is minus two minus to plus five, which is three a minus one plus three, which is to which we're gonna call V. This outlines a parallelogram whose area is given by, uh, you across the size of the crispy. The area of the triangle is less half of this. So let's calculate U cross V using the determinant method of I J k u is 102 on D. V is minus 232 For this I times zero to 32 minus J A at homes The determinant one to minus 22 plus k times the determinant off 10 minus 23 This determinant is minus six I. Then we'll have to plus four, which is six so minus six j and then we'll have plus three. Okay, The size of this vector is minus six squared plus minus six squared plus three squared. This is 36 plus 36 plus nine. Um, which gives ah, 81 square it which is nine. The area is half of this, which is nine over to

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