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Find the volume of the parallelepiped with adjacent edges $ PQ, PR $, and $ PS $.

$ P (3, 0, 1) $ , $ Q (-1, 2, 5) $ , $ R (5, 1, -1) $ , $ S (0, 4, 2) $

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16 units $^{3}$

02:54

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Aruzhan H.

February 16, 2022

Find the area of the triangle with vertices (0,0,0),(?2,?2,2), ( 0 , 0 , 0 ) , ( ? 2 , ? 2 , 2 ) , and (?2,?1,0). ( ? 2 , ? 1 , 0 ) .

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

02:51

Find the volume of the par…

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01:04

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03:51

Hello. Welcome to this. Les. Lend this lesson. We have the edges of the parallel pain. A speak e l p L P s and you find in the volumes the volume of it. So the volume is giving us the triple. For that. It is a PTL dot p l across P s. Okay, so this requires us to find a P kill. That is the 21 minus three to minus zero five minus one. That becomes negative. Four, two, and four. Okay, The next one is to find P. How p l. And that is five minus three one minus zero. Negative. One minus one. Yeah. So that is to one negative two. Yeah. The next step is to find a P s. So p s is Yeah. Zero. My necessary for my necessary two minus one. And that is negative. Story four and one. Okay, so let's find, uh, volume of the parallel pep given us this. Okay? Well, yeah, yeah. So here we have the increase of the first one. Mostly, uh, if we have the cross product, then we have the determinant of the Metris beginning work there. Unit vectors. But now we have triple product. So it begins with the first victor, then the second. And the third factors are the interests of the determinant mattress. So let's spend with the first role as a co Fatima trees. So he would lead the first ruin. A second column of 14 Negative. 21 minus two. Then we have to negative story. Negative. Two and one. That is. We delete this, we delete that. Okay, The next one is four. This four. So delete the whole of the roll it the router has the for the column has the fall so that we have two negative. 314 Mhm. Okay, so this is negative for out. And this is one. Yeah, last eight. Okay, so you have to minus six. The north plus four, eight plus three. Okay, so here we have negative. Four times. Nine. That is negative. 36 than negative. Two negative. Two times negative. Four. So the less a positive eight. Yeah. Then we have negative. We have a positive four times 11. So that is 44 and all of them gives us. So we add, they stood that McCain 50. Yeah, to the new subtract practices from it, I guess, Is mhm 16. Okay, so we have sustain mhm cubic units. Yeah, as the volume of the parallel pain architecture time. That's the end of the lesson.

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