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Prove that$ (a \times b) \cdot (c \times d) = \left \| \begin{array}{ll} a \cdot c & \mbox{$ b \cdot c $}\\ a \cdot d & \mbox{$ b \cdot d $} \end{array} \right \| $

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03:30

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Johns Hopkins University

Campbell University

Harvey Mudd College

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, or where we're trying to find the dot product of across B and c. Cross D. And figuring out what this is equal to ideally as a determinant. And the way that we can approach this is by taking see crusty and by writing it as a new vector. Yeah, so we're really looking at a cross B dot e and our cross product identities tells us that this is the same thing as a dot. Be cross E. If we expand this out, this is just a dot, be cross. And I remember E was just see cross D at this point, we can use the helpful identity that we've used in the past that says that a triple cross product is really The 1st, 3rd time, 2nd minus first at second time. Third, let's put that into use. This is a diet. And then, like I said, first times first dot third, that will be he got the time, see minus 1st and 2nd, that will be e dot c. Time's D. Now you'll notice that be dot de is just a number as his be dot C. Therefore, this is really a dot, a number of times see -1. Our Number of Times D. What we're really looking at is a dot c, times this number be dot de minus the number. Be dot c times a dot be. You will notice that this looks kind of like a determinant, meaning we could write this as the determinant of the matrix A dot c. Pierotti. No. And you don't see a dot de. So that we're looking at a dot c b dot de minus b dot c a dot de. And that is what this dot product of cross products is equal to control.

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