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Vectors in Space

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.

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Vectors in Space

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

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

  3. 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)

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

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

  6. 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.

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

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

  9. 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

  10. Direction Angles

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

  12. Property of Direction Cosines • If a, b, and g are the direction angles of a nonzero vector v in space, then • cos2a + cos2b + cos2g = 1

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