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

$ a = \frac{1}{2} i + \frac{1}{3} j + \frac{1}{4} k $ , $ b = i + 2j - 3k $

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03:50

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Johns Hopkins University

Missouri State 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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02:33

mm. Let's try another cross product problem. This time with fractions. We want to look at the cross product of a time. A cross B. Where a is the vector one half I Plus 1 3rd J Plus 1 4th K. Let's write that in our matrix And B is one I plus two j -3K. Let's write that on our matrix as well. Then the cross product A cross be. We can use the formula from our textbook where we ignore the first column and look at one third times minus three -1 4th Times two. One third times minus three minus 1/4 times two I minus. And then we ignore the second column And look at 1/2 times -3 Times 1 4th -1. Yeah, of one half times minus three minus one. Fourth types one jay. And lastly we add to that, we ignore the third column And look at 1/2 times two -1 3rd Times one. 1/2 times two minus one third times one. Okay simplifying all of us a little bit. One third times minus three is minus 3/3 Or just -1 And 1 4th times two is to over four or one half -1 -1 half. I minus negative three times one half is minus 3/2 In 1 4th times one is just 1/4 so negative 3/2 -1/4 J plus one half times two is 2/2 or just one minus 1/3 times one is just 1/3. Okay. Simplifying this even more We have negative 1 -1 half. That's -3/2. I and then negative 3/2 minus 1/4. That's the same thing as minus 6/4 minus 1/4 Which would give us -7/4 but we're subtracting that so we're gonna say plus 7/4 jay plus one minus one third. That's three thirds minus one third or two thirds. Okay? Or we can write this as a vector -3/2, 7/4 to over three. Thanks for watching.

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