We live in a Three Dimensional World Vectors in Space We live in a Three Dimensional World

Rectangular Coordinates in Space Right handed coordinate system We now have an ordered triple (x,y,z) associated with each point. Graphing examples

Representing Vectors in Space Since we have a new axes, we will now need a third unit vector to represent the z axis. i = (1, 0, 0) j = (0, 1, 0) and k = (0, 0, 1)

Position Vector To find the position vector, we will now have v = (a2 – a1)i + (b2 – b1)j + (c2 – c1)k

Addition, Subtraction and Scalar Multiplication All rules that applied in two dimensions, now apply in three dimensions

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

Dot Product We find the dot product the same way we found it in two dimensions, we just add the third dimension

Angle Between Two Vectors We use the same formula we used in two dimensions including the third dimension

Direction Angles of Vectors in Space This is the only truly new operation. There are three direction angles a = angle between v and the positive x-axis, 0 ≤ a ≤ p b = angle between v and the positive y-axis, 0 ≤ b ≤ p g = angle between v and the positive z-axis, 0 ≤ g ≤ p

Direction Angles

Direction Cosines The direction cosines play the same role in space as slope does in the plane.

Property of Direction Cosines If a, b, and g are the direction angles of a nonzero vector v in space, then cos2 a + cos2b + cos2 g = 1