Michael Corral: Vector Calculus

14 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE
We can now easily prove Theorem 1.1 from the previous section. The distance d
between two points P=(x 1 ,y 1 ,z 1 ) and Q=(x 2 ,y 2 ,z 2 ) in 3 is the same as the length
of the vector w−v, where the vectors v and w are defined as v=(x 1 ,y 1 ,z 1 ) and w=
(x 2 ,y 2 ,z 2 ) (see Figure 1.2.8). So since w−v=(x 2 − x 1 ,y 2 −y 1 ,z 2 −z 1 ), then d=‖w−v‖=
√
(x2 − x 1 ) 2 +(y 2 −y 1 ) 2 +(z 2 −z 1 ) 2 by Theorem 1.2.
x
0
z
v
P(x 1 ,y 1 ,z 1 )
w
w−v
Q(x 2 ,y 2 ,z 2 )
y
Figure 1.2.8 Proof of Theorem 1.2: d=‖w−v‖
A
☛ ✟
✡Exercises
✠
1. Let v=(−1,5,−2) and w=(3,1,1).
(a) Find v−w. (b) Find v+w. (c) Find v
‖v‖ . (d) Find∥ ∥ ∥
1
2 (v−w)∥ ∥ ∥.
(e) Find ∥ ∥ ∥
1
2 (v+w)∥ ∥ ∥. (f) Find−2v+4w. (g) Find v−2w.
(h) Find the vector u such that u+v+w=i.
(i) Find the vector u such that u+v+w=2j+k.
(j) Is there a scalar m such that m(v+2w)=k? If so, find it.
2. For the vectors v and w from Exercise 1, is‖v−w‖=‖v‖−‖w‖? If not, which
quantity is larger?
3. For the vectors v and w from Exercise 1, is‖v+w‖=‖v‖+‖w‖? If not, which
quantity is larger?
B
4. Prove Theorem 1.5(f) for 3 . 5. Prove Theorem 1.5(g) for 3 .
C
6. We know that every vector in 3 can be written as a scalar combination of the
vectors i, j, and k. Can every vector in 3 be written as a scalar combination of
just i and j, i.e. for any vector v in 3 , are there scalars m, n such that v=mi+nj?
Justify your answer.