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If $ a = \langle 1, 0, 1 \rangle $, $ b = \langle 2, 1, -1 \rangle $, and $ c = \langle 0, 1, 3 \rangle $, show that $ a \times (b \times c) \neq (a \times b) \times c $.

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Perform the calculations in the ( ) first to show that the final results are notequal.

04:24

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Johns Hopkins University

Oregon State University

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 cross product problem where we will be looking at the cross product of a cross B. Cross C. Given the vector eight 101 B 2, -1. And see 013 1st. We need to find a Crosby using the same technique from our textbook that's going to be zero times negative one -1 times one zero minus one. And then we'll be looking at the negative version of one times negative one. That'll be negative one minus two. And lastly when we ignore K Looking at one time is one is 1 zero. And so when we put this all together, that becomes the new vector negative one, negative, negative three and 1 0 is just what? And so this is going to go right in here Giving us a negative one three one. And now we're looking for the cross product of -1, and 013. Let's use the same technique. We'll ignore our first column three times three -1 times one. That's going to be 9 -1 minus now. We'll ignore the second column and we look at negative three minus zero. And lastly we'll ignore our third column. Well look at -1 times one minus three times zero zero. So a Crosby Cross C all together is eight negative negative three is three negative one. Let's remember this number because what we're gonna do now is try and do the same thing but with a different order of operations. What is a cross be crazy? Is it the same? Well let's find out Let's start with the parentheses. Be cross see remember that's 2 1 -1 Across 013. We ignore our first column, that's three minus negative one. Just three plus one- for Middle one. two times 3 is six minus zero 60 And lastly when we and our K2 minus zero, simplifying all of that. That gives us four negative six two. So we're going to put that right in here. Be cross C is for negative six two. And now let's try the math again. Using the same technique from our textbook. If we ignore our first column zero times two -1 times negative six and zero minus negative six or 0-plus six yeah We subtract our Middle one When we're looking at two times 1 -4 times one two minutes four. And then when we ignore a third column that leaves us with negative six minus zero, not necessarily good. When we simplify all of this that gives us six 2 -4 is negative two negative negative too and negative six. So it turns out that a cross be cross C. Is not equal to a cross B chrissie the order in which we do these calculations matters. Thanks for watching

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