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Suppose that $ a \neq 0 $.(a) If $ a \cdot b = a \cdot c $, does it follow that $ b = c $?(b) If $ a \times b = a \times c $, does it follow that $ b = c $?(c) If $ a \cdot b = a \cdot c $ and $ a \times b = a \times c $, does it follow that $ b = c $?

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(a) No(b) No(c) Yes

02:58

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

University of Michigan - Ann Arbor

University of Nottingham

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.

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Welcome back to another dot and cross products. Problem. Yeah. In this case we're trying to figure out if a dot B equals dot C, But a is not equal to zero. Does that mean at B is equal to see? Well, the answer actually is no. And we can see this in a couple different ways. First, let's imagine A B is equal to zero. That means that A and be are orthogonal, but a C is another vector orthogonal to a. Then a dot C is also going to be zero. And so we could have, the left side is equal to the right side. Even though B and C are clearly different vectors, we can see this a little more mathematically if we rearrange this equation. So it says a dot be minus a dot C equals zero. The mattress tells us that a diet b minus C equals zero and therefore B minus c. Just has to be perpendicular to A but it doesn't say that B has to equal. See. All right, what about if a cross B equals a cross C. Well, same idea. Yes. Let's say both sides are equal to zero again, that means that A and B are parallel and the D n C are also parallel. We had A B and see a Crosby would be zero. A Cross C would be zero, but B and C are very clearly different. Once again, we can see this by looking at a cross, B minus A cross C equals zero. And we know that since this distributes as well we have a cross B minus C equals zero, which just tells us that B minus C has to be parallel to A. But what happens if both this is true, A dot B equals C and this is true, A Crosby equals a cross C. What does that tell us about BNC this time? Well, just as before, we require B minus C to be perpendicular to A. But we also require B minus C to be parallel to A. And the only vector that's both perpendicular and parallel to another vector has to be the zero vector. Therefore b minus C has to be the zero vector, or equivalently B has to equal. See Thank.

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