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If $ a + b + c = 0 $, show that$$ a \times b = b \times c = c \times a $$

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02:14

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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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Welcome back to another cross product problem where we're trying to show if the sum of vectors A, B and C equals zero, that means a cross B equals B. Cross C equals C. Cross A. How do we get there? Well, if A plus B plus C equals zero, that tells us that A equals negative B minus C. Therefore a cross B is really negative B minus C. Crosby and cross products distribute. And so we have this is negative B. Crosby minus C. Crosby. But the cross product of any two parallel vectors, It's going to be zero minus. See Crosby. And another thing we know is that negative, be crusty? Is the same thing as positive be cross C. So there's our first equality. Next thing we can do is the same idea. But we'll start with writing B equals negative A minus C. And then we can look at B Cross C be crushed. C is then going to be negative A minus C. Chrissie. Again, this distributes, so we're looking at negative A cross C minus C cross C. Once again negative A. Craftsy. It's the same thing as C cross A. And any vector cross product with itself or any vector parallel to it zero. And therefore we have shown The 2nd equality as well. Thanks for watching.

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