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Find the area of the parallelogram with vertices $ A (-3, 0), B (-1, 3), C (5, 2) $, and $ D (3, -1) $.

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20 units squared.

03:27

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Johns Hopkins University

Oregon State University

Harvey Mudd College

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.

01:59

Find the area of the paral…

00:53

07:07

Verify that the points are…

02:39

03:31

Find the area of the quadr…

03:30

03:56

Let's try a parallelogram problem where we're trying to find the area of a parallelogram defined by point A. At -30. We'll call this one a B is negative 1 3 or happy See is that 5 to? We'll call that C. & D. is at three negative one. We'll call that D. In order to find the area of this parallelogram, we need to find vectors A. B. And A D. For example, that we can use to find the area of the entire parallelogram. So A. B. That's just going to be b minus A. Or negative one minus negative three. That'll be two, three minus zero. That'll be three. And since we'll be using the cross product here, the 3rd coordinate is just zero. Similarly, we can find the vector A. D. Which is just d minus a three minus negative three. His six -1 0 is -1. And once again our last coordinate will be zero. Since we're trying to find the cross product A cross B. Let's go ahead and put these vectors and our matrix here. And then we can use the technique from the textbook in order to find this cross product specifically, we can ignore the first coordinate And then look at three times 0 zero times negative one. That's just going to be 0 0. I when we ignore the second column, two times 0 zero times 6 began going to be zero. Rain is zero jay. And lastly, when we ignore the third column two times negative, one -3 times six -3 times six. Okay, writing this all out as one vector That gives us zero I minus zero J, two times negative one is negative two minus 18. That's minus 20. Okay, now the the area of a parallelogram is going to be the magnitude of the vector A crossed B that we just calculated. And since we have no components in the X direction, no components in the Y direction magnitude is just going to be the absolute value of negative 20 which is Just 20 units squared. Thanks for watching.

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