Vectors, Part 5

Vectors in Two and Three Dimensions

Part 5: Vectors in Space

We think of three-dimensional space by starting with a horizontal xy-plane and extending upwards and downwards using a third coordinate, z. In this 3D space, we can create vectors just as we did in the plane.

Vectors in space may be described by ordered triples of coordinates (a, b, c). Geometrically, the vector so defined can be thought of as an arrow pointing from the origin to this point in space.

As with vectors in the plane, we define addition and subtraction coordinatewise:

(a,b,c) + (d,e,f) = (a+d,b+e,c+f)

(a,b,c) - (d,e,f) = (a-d,b-e,c-f)

Compute the sum and difference of (1,2,-3) and (-7,1,0). Graph these vectors, and their sum and difference, both on paper and in your worksheet.

Vectors in space also have direction and length. These can be computed in the same way as for 2D vectors, taking the additional coordinate into account in our formulas. Thus, the magnitude or length of a vector v = (a,b,c) is

and the dot product of two vectors v = (a,b,c) and w = (d,e,f) is

<v,w> = ad + be + cf.

Just as before, the dot product tells us about the relative direction of two vectors.

Express the cosine of the angle between vectors v and w in terms of their dot product and their lengths. (Hint: Review your 2D calculation in Part 2, and determine what changes when each vector has a third component.)

Compute the dot product of (1,2,-3) and (-7,1,0), and calculate the angle between them.
Plot the vectors (-2,3,1) and (1,5,3) in your worksheet, and calculate the angle between them.

In the plane, we saw that we could express a vector as the sum of a multiple of the horizontal unit vector, i, and a multiple of the vertical unit vector, j. We can do the same in space but we need three unit vectors now -- the same two vectors i = (1,0,0) and j = (0,1,0) that lie in the xy-plane and a new vector k = (0,0,1) perpendicular to this horizontal plane.

Express the vector (-1,5,2) as a sum of scalar multiples of i, j, and k.

Multiplication of vectors in space has more possibilities than in the plane. As we shall see in Part 7, there is a second notion of multiplication for vectors in space that has geometric significance. In particular, it expresses the relationship of the vectors i, j, and k rather nicely.