By David Massey

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**Additional resources for Worldwide Multivariable Calculus**

**Example text**

Thus, its direction, the unique vector in the direction of d is 1 d = p ( 4, |d| 21 2, 1). Note that the direction is unitless, the meters cancel out. 15. , dist(p, q) = |q p| = |p q|. For many calculations, it is simpler to deal with this unit vector as we have written it, rather than multiplying the scalar times each component. 24 CHAPTER 1. 16. 5, 2, 0, 3), (8, 12, 1, 7). For parallel vectors, decide if they point in the same or opposite directions. Solution: We need to see which vectors are (necessarily, non-zero) scalar multiples of other vectors, and also note whether the (possible) scalars are positive or negative.

16. 5, 2, 0, 3), (8, 12, 1, 7). For parallel vectors, decide if they point in the same or opposite directions. Solution: We need to see which vectors are (necessarily, non-zero) scalar multiples of other vectors, and also note whether the (possible) scalars are positive or negative. First, note that the first three vectors all have a 0 for their third component, while the fourth vector has a 1 in its third component. Thus, the fourth vector cannot possibly be a scalar multiple of any of the first three.

For parallel vectors, decide if they point in the same or opposite directions. Solution: We need to see which vectors are (necessarily, non-zero) scalar multiples of other vectors, and also note whether the (possible) scalars are positive or negative. First, note that the first three vectors all have a 0 for their third component, while the fourth vector has a 1 in its third component. Thus, the fourth vector cannot possibly be a scalar multiple of any of the first three. So, now we ask: is there a scalar r such that (2, 3, 0, 4) = r( 3, 4, 0, 6)?