Chapter 11
11.1 3D Coordinate System
A. Obj: to plot points in space; to find the distance in space; to write an equation of a sphere.
B. Facts:
1. 3D coordinate system
2. 3 planes divide the system into 8 Octants. Octant I is where all coordinates are positive.
3. distance formula in space:
4. Midpt. Formula in space:
5. Equation of a sphere:
Radius:
11.2 Vectors in Space
A. OBJ: Find the component forms of the unit vectors in
the same direction of, the magnitudes of, the dot products of, and the angles between vectors in space; Determine whether vectors in space are
parallel.
B. Facts:
1. Component form of a vector in space:
v =v1, v2, v3.
2. Unit vector form where i= 1, 0, 0, j = 0, 1, 0, andk = 0, 0, 1:
v = v1i + v2j + v3k
3. Component form from P (initial pt.) to Q (terminal pt.):
v = v1, v2, v3
= q1 – p1, q2 – p2, q3 – p3.
4. Rules for vectors in space:
5. Angle between 2 non-zero vectors (0<θ<π) can be found using dot product:
Reminder: if dot product = 0, then the vectors are orthogonal.
6. Parallel vectors: the vectors u, v, and w are parallel because u =2v and
w = –v.
11.3 Cross Product of 2 Vectors
A. OBJ: Find cross products of vectors in space; Use geometric properties of cross products of vectors in space; Use triple scalar products to find volumes of parallelepipeds.
B. FACTS:
1. Cross product (3d vectors only):
2. Properties of Cross Product:
3. Triple Scalar Product:
11.4 Lines and Planes in Space
A. OBJ: Find parametric and symmetric equations of lines in space. Find equations of planes in space. Sketch planes in space. Find distances between points and planes in
space.
B. FACTS:
1. Lines in space through a pt. P on a line and parallel to vector V:
Symmetric Equation of a line in Space:
2. Planes in Space through a pt. P in a plane and normal to vector V:
This plane consists of all points Q(x, y, z) for which the vector is orthogonal to n.
Using the dot product, you can write:
n = 0
a, b, cx – x1, y – y1, z – z1= 0
a(x – x1) + b(y – y1) + c(z – z1) = 0(standard form)
ax + by + cz + d = 0(general form)
3. 2 planes in space are either parallel or intersecting at a line. To find the angle between the planes:
*They are perpendicular when n1n2 = 0 and parallel when n1 is a scalar multiple of n2.
4. Distance between a pt. P and a plane :