Find the determinants in Exercises 3540, assuming that 3abcdefghi 3 4

MATH241­Lecture 1: 3d Coordinates 3  3D coordinate system is denoted by R  The distance between two lines in three dimensions is given by 2 2 2 D= (√−x ) +1y−y ) +(z1z ) 1 2 3  We need to be careful about translating things from R to R because things are not always what they seem. x=3 R1 R2 R 3 Ex: Plot in , and In R , it’s just a point on a number line. 2 In R , you get a vertical line. R3 And in , we get a plane. As we can see, things are very different and not at all intuitive. Equations of Lines Let us now look at the equation of a line in 3D. The usual equation for a line in 2D, y=mx+b , actually describes a plane in 3D! So we need to use another way to get what we are looking for. That is where vector equations come in. A vector function is a function that takes in two or more variables and returns a vector. Let’s consider the following vector function. r(t)=(t ,5) Let’s plug in a few values for t and see what we get. T=0: (0,5) T=3: (3,5) T=2: (2,5) This looks familiar! Yes indeed! This is the line y=5 in 2D! Vector Equation of a Line We have seen above what vector equations can give us. Now we are ready to formally introduc