10.6Vectors in Space

z

y

x

2

-2

2

-22

-2

O

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

Find the distance from P1 = (2, 3, 0) to P2 = (-1, 3, -2).

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

P = (a, b, c)

v = ai + bj + ck

Position Vector

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

v = (x1- x2)i+(y1 - y2)j+ (z1 - z2)k

P2 = (x2, y2, z2).P1= (x1, y1, z1),

Find the position vector of the vector v=

P1P2 if P1= (0, 2, -1) and P2 = (-2, 3,-1).

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

Theorem Unit Vector in Direction of vFor any nonzero vector v, the vector

is a unit vector that has the same direction as v.

Find the unit vector in the same direction as v = 3i+ 5j - k .

Theorem Properties of Dot ProductIf u, v, and w are vectors, thenCommutative Property

Distributive Property

Theorem Angle between Vectors

Theorem Direction Angles

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

Theorem Property of Direction Cosines

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