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(a) Show that $ i \cdot j = j \cdot k = k \cdot i = 0 $(b) Show that $ i \cdot i = j \cdot j = k \cdot k = 1 $

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a) $i \perp j, j \perp k,$ and $k \perp i$b) $0+0+1=1$

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

Campbell University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

01:23

(a) Show that $i \cdot j=j…

00:18

$$\text { Show that: }\lef…

Okay, so I'm going to show that we can write I as the as a vector one zero zero j as the vector zero one zero and chaos the vector zero zero one. All right, so now it's just a matter of doing these dot products. So let's just look at one and then I'll let you kind of figure out the rest, right? So if we're going to do one time zero plus zero times one plus, you're in time zero, we always see that if these two vectors are different, then we're always going to match up a zero with something else and always get zero. Okay, so this makes sense. And if we look at these guys here, would you I with eye, we're always going to get the two components with the ones to match up and in the zeros with zero. So we always get one. Okay, that's the same with the J and the cakes of the ones. We're just in different places.

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