Wed. Apr. 15– Calculus Lecture #23.2
Vectors, Dot Products, Projections, and Planes
11.1.9)
If v and w are unit vectors, what is the geometrical meaning of v·w? What is the
geometrical meaning of (v·w)w? Draw a figure with v = i and w = (3/5)i +(4/5)j.
(based on 11.2.20)
A = (1, –1, 0), B = (4, 2, 4).
Find the projection P of B along (parallel to) A.
Find the projection of B perpendicular to A.
11.1.16(a)
Draw the unit vectors u1 = (cos q, sin q) and u2 = (cos f, sin f). By dot products,
find the formula for cos(q – f).
11.1.45) A methane molecule has a carbon atom at (0, 0, 0) and hydrogen
atoms at (1, 1, –1), (1, –1, 1), (–1, 1, 1), and (–1, –1, –1). Find
(a) the distance between the hydrogen atoms
(b) the angle between the vectors going out from the carbon atom to the
hydrogen atom.
11.2.8) Find the equation of the plane through
P0 = (1, 2, –1) perpendicular to N = i + 2j – k.
11.2.2) Find two points P and P0 on the plane
x + 2y + 3z = 6 and a normal vector N.
Verify that N·(P – P0) = 0.
11.2.44) From their equations, how can
you tell if two planes are (a) parallel
(b) perpendicular (c) at a 45o angle?
use planes
ax + by + cz = d
Ex + Fy + Gz = H
11.2.9) Find the equation of the plane through (1, 0, 1) parallel to x + 2y + z = 0.
(based on 11.2.36) What is the distance between the planes x + y + 5z = 7 and
3x + 2y + z = 1?
11.2.31) How far is the plane x + y – z = 1 from (0, 0, 0) and also from
(1, 1, –1)? Find the nearest points.
11.1.31) In the semicircle find C and D
in terms of A and B. Prove that C·D = 0
(they meet at right angles).