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Use Exercise 50 to prove that$$ a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0 $$

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02:54

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Missouri State University

Campbell University

Oregon State University

Idaho State University

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.

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Yes, welcome back to another cross product problem. Our last proof, we showed that across peak rossi is equal to a dot C times b minus a dot B times C. This time we're going to use that to evaluate a cross the cross C plus B. Cross secrecy plus C. Cross a Crosby and see what that's all equal to. So, using this identity, we can start to evaluate these terms by term by term a crispy crust. See, has already written on the page, it's a dot C times b minus a dot be time see plus. Then we're looking at B cross, see, cross A. Looking at the first in third terms here, that will be be dot A time see minus. And then the 1st and 2nd terms here, that will be be dot C times A plus. And then same idea. We're looking at the First and 3rd terms in the dark product will be c dot be times a minus. And in the 1st and 2nd terms c dot A times B. Yeah. Well, notice that we've got an A dot C, times B and a minus C dot A times B since a dot C and see that they are the same thing those are going to cancel. We've also got a B dot A times C right here and we've got a negative a tubby time. See right here again, A dot B and P dot A are the same things. Those canceled. And lastly, sea dot B times a negative beat at C times A. There's going to cancel as well. And we're left with A big old zero. The sum of all of these triple cross products is just zero. All right. Thank you for watching.

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