Sec 13.1 The Three-Dimensional Coordinate System

1. z y O x Sec 13.1 The Three-Dimensional Coordinate System There are threecoordinate planes: xy-plane, xz-plane, and yz-plane. These three planes separate three-space into 8 octants.

2. Coordinates in Three-Space: The coordinates of a point P in three-space are (x,y, z) where x is its directed distance from the yz-plane; y is its directed distance from the xz-plane; z is its directed distance from the xy-plane.

3. The Distance Formula: The distance between two points P(x1,y1,z1) and Q(x2,y2,z2) is given by

4. The Midpoint Formula: The coordinates of the midpoint of the line segment joining two point P(x1,y1,z1) and Q(x2,y2,z2) are:

5. Equation of a Sphere:An equation of a sphere of radius r, centered at C(h, k, l) is given by In particular, if the center is the origin O, then an equation of the sphere is

6. Sec 13.2 VECTORS A scalar quantity can be characterized by a single real number. Examples: area, volume, time, mass, temperature. A vector is a quantity involving both magnitude and direction and cannot be characterized completely by a single real number. Examples: force, velocity, acceleration.

7. Geometric Representations A directed line segment is used to represent a vector quantity. A directed line segment with initialpoint(tail), P, and terminal point(head), Q, is denoted by Its length (or magnitude) is denoted by Q P

8. Equivalent Vectors:Two vectors are considered to be equivalent if the have the same magnitude (length) and direction. • Definition of Vector Addition: • To find the sum, (or resultant), of two vectors u and v, • move v until its tail coincides with the head of u; then u + v is the vector connecting the tail of u to the head of v. • move v so that its tail coincides with that of u; then u + v is the vector with this common tail and coinciding with the diagonal of the parallelogram that has u and v as sides. • (This is called the Parallelogram Law)

9. Scalar Multiplication: The scalar multiple of the vector u is given by cu (where c is any real number) which is a vector with length |c| times that of u and is similarly or oppositely directed depending on whether c is positive or negative. In particular, (−1)u (usually written −u) has the same length has u, but opposite direction. This is called the negative of u because when it is added to u, the result is just a point, called the zero vector (denoted by 0).

10. Properties of Addition:u + v = v + u (u + v) + w = u + (v + w) Identity Vector for Addition: The zero vector, 0, is the identity vector for addition because u + 0 = 0 + u = u. Subtraction of vectors is defined by u − v = u + (−v).

11. Algebraic representation of vectors: If the initial point of a vector a is at the origin O of a rectangular coordinate system, then the terminal point of a has coordinatesin a two- or three-dimensional coordinate system. These coordinates are called the components of vector a, Notation: We write We call the position vector of the point

12. Algebraic representation of vectors: The magnitude or length of a vector a is denoted by

13. Adding and Subtracting Vectors:

14. Properties of Vectors: If a, b, and c are vectors inand α and β are scalars, then 1. a + b = b + a 2. a + (b + c) = (a + b) + c 3. a + 0 = a 4. a + (- a) = 0 • α (a + b) = αa + αb • (α + β) a = αa + βa • (αβ) a = α (βa) 8. 1 a = a

15. The Standard Basis Vectors:

16. Unit Vectors: A unit vector is a vector whose length is 1 unit. For examples, the standard basis vectors are all unit vectors. In general, if a ≠ 0, then the unit vector that has the same direction as a is