1. 12.10 Vectors in Space

2. Vectors in three dimensions can be considered from both a geometric and an algebraic approach. Most of the characteristics are extensions from vectors in the plane. The three basic vectors are α, β, and γ are the direction angles measured from the x, y, and z axes.

3. Vector Relationships:

4. Ex 1) a) b) c)

5. More similarities with vectors in the plane: Ex 2) Determine the angle between vectors

6. Q(x, y, z) and P(a, b, c) Vector Equation of a Line (in space): and therefore the parametric equations are: x = a + tv 1 y = b + tv 2 z = c + tv 3 Ex 3) Determine both vector & parametric equations of the line in space determined by P(1, 4, 7) and R(–6, –4, –8) direction vector: vector eqtn: parametric eqtns:

7. Normal vector: this is a vector perpendicular to a plane – meaning the vector is perpendicular to all the vectors that contains a specific point in the plane. Ex 4) Determine an equation of the plane containing the point P(1, 2, 3) with normal vector Let Q(x, y, z) represent all points in the plane

8. An operation that applies to three-dimensional vectors, but not two- dimensional is the cross-product. We will use determinants to help us solve. Remember: vector (not a scalar) This cross-product is perpendicular to the plane that the two vectors make. This fact enables us to find the equation of that plane. You will be given 3 points in a plane. Use them to create two vectors. Find the normal to the plane using those two vectors. Then create the equation of the plane using any of the three points.

9. Ex 5) Find an equation of the plane containing the points P(2, –1, 6), Q(0, 1, 5) and R(4, –1, 2). (1) Determine vectors (2)

10. Ex 5) cont… (3) equation of plane using P or

11. Homework #1210 Pg 664 #1–13 odd, 19, 22, 25, 27, 31, 32, 35, 37