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Find the cross product $ a \times b $ and verify that it is orthogonal to both $ a $ and $ b $.

$ a = \langle 4, 3, -2 \rangle $ , $ b = \langle 2, -1, 1 \rangle $

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$\mathbf{a} \times \mathbf{b}=\mathbf{i}-8 \mathbf{j}-10 \mathbf{k}$

02:39

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Johns Hopkins University

Campbell University

Baylor 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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Let's try another cross product question. We're looking at the cross product of A. Which is 43 -2. Let's write that in the second row of our matrix negative two and B. Two negative one negative one. Sorry to negative one positive one. And then let's use the formula provided in the text book. So first we ignore our first column And look at the product three times 1 minus negative two times negative one. That's going to give us three times 1 minus The product of -2 and negative one I minus. Next we'll ignore J. And we'll look at four times 1 -20 Times two. That's four times one minus negative two times two jay. And lastly we'll ignore the third column And we'll look at four times negative one -3 Times two. Is us four times a negative one -3 Times two. Okay if we simplify all of this, that gives us the three times one is three -2 times negative one is 2. It's really an 3 -2. I minus four times 1 is four -2 times two is -4. So we're looking at four minus native for jay plus And we have four times negative one. It's minus four -3 times two minus six. Okay? And so putting this all together we get 3 -2 is one. I we can just straight I four minus negative four is eight, so we're minus eight J And then -4 -6 is -10. Okay? Alternately We can write that as one minus eight -10. All as a vector. Thanks for watching.

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