1. 8.6 Vectors in Space

2. z 2 -2 -2 O y 2 2 -2 x

3. Theorem Distance Formula in Space
If P1 = (x1, y1, z1) and P2 = (x2, y2, z2) are two points in space, the distance d from P1 to P2 is

4. Find the distance from P1 = (2, 3, 0) to

5. If v is a vector with initial point at the origin O and terminal point at P = (a, b, c), then we can represent v in terms of the vectors i, j, and k as
v = ai + bj + ck

6. Position Vector
P = (a, b, c)
v = ai + bj + ck

7. Theorem
Suppose that v is a vector with initial point P1= (x1, y1, z1), not necessarily the origin, and terminal point P2 = (x2, y2, z2). If v = P1P2 then v is equal to the position vector
v = (x1- x2)i+(y1 - y2)j+ (z1 - z2)k

8. P1= (x1, y1, z1),
P2 = (x2, y2, z2).
v = (x1- x2)i+(y1 - y2)j+ (z1 - z2)k

9. Find the position vector of the vector v= P1P2 if P1= (0, 2, -1) and P2 = (-2, 3,-1).
v = (-2 - 0)i+ (3 - 2)j+[-1-(-1)]k = -2i + j

11. If v = -2i+ 3j + 4k and w = 3i+ 5j - k find
(a) v + w (b) v - w
(a)
v + w= (-2i+ 3j + 4k) + (3i+ 5j - k)
= (-2+3)i + (3+5)j + (4-1)k
= i + 8j + 3k
(b)
v - w= (-2i+ 3j + 4k) - (3i+ 5j - k)
= (-2-3)i + (3-5)j + (4-(-1))k
= -5i -2j + 5k

12. For any nonzero vector v, the vector
Theorem Unit Vector in Direction of v
For any nonzero vector v, the vector
is a unit vector that has the same direction as v.

13. Find the unit vector in the same direction as w = 3i+ 5j - k .
First we will find the magnitude of v.

16. Theorem Properties of Dot Product
If u, v, and w are vectors, then
Commutative Property
Distributive Property

17. Theorem Angle between Vectors

19. We find that

20. Theorem Direction Angles

21. Find the direction angles of v= -3i+2j-k.

22. Theorem Property of Direction Cosines

23. A nonzero vector v in space can be written in terms of its magnitude and direction cosines as