Merced College; Don Power
LINEAR ALGEBRA - CHAPTER 3, LECTURE
3.1 Introduction to Vectors (Geometric) Hwk: 1af, 2bgh, 3def, 5b, 6d, 7, 8, 9, 11, 12, 14, 15, 16, 17, 18, T1
vectors as directed line segments (arrows): initial point, terminal point
notation: bold face (in print) or with arrow over top (handwritten)
Ex: v
= 
sum, difference, equality, scalar multiples
geometrically
using coordinates, with v = (v1,v2) in this text { or <v1,v2> other texts}
extension to 3-space
geometrically: right-hand coordinate system
using coordinates
Calculation of 
from the coordinates of the points in R3
Translation of axes to (x',y'):
If coords of translated origin are (k,l), then translation equations are x'=x-k, y'=y-k
From exercises:
Find midpoint (or point some fraction of the way)along line segment connecting two points
See #8,9: given vectors u, v, w, x, find scalars c1, c2, c3 (or show they don't exist)
such that c1u+c2v+c3w=x
3.2 Norm of a Vector; Vector Arithmetic Hwk: 1bce, 2c, 3ce, 4, 5a, 7, 8bd(for parts e, g), 9ac, 10, 11, 12 (hint: the name of this inequality is the "triangle inequality"), 14 (for part d only; do the proof for a vector in 3 dimensions)
Properties of vector operations: basically the same as properties of "vector spaces" in 5.1:
closure + ´
associative + ´
identity + ´
inverse +
commutative +
distributive across 2 scalars or 2 vectors
The message later: other algebraic structures (besides vectors) also behave this way
Proofs may be geometric or analytic; geometric proofs work only for R2
Norm of a vector is its length.
Notation ||u||
Formula: ||u|| = sqrt (u12 = u22 + u32) for R3: proof by Pythag thm
Unit vector: A vector with norm = 1
Distance between two points is the norm of the vector linking
them: ||||
Ex
Note: ||ku|| = |k| ||u||
Ex find norm of vector (-4,-8,12)
For a fuller discussion, go to http://www.mcasco.com/p1va.html
3.3 Dot Product; Projections Hwk: 1ac, 2ac, 3abc, 4ac, 5ac, 6ac, 8ab (think of slope), 9b, 12 (find all 3 angles), 14, 16b, 20 (write two vectors that define the angle), 21ab, 25, 27, 28, 29, 31
"Euclidean Inner Product"
Def:
cosine formula
Calculation of u dot v in terms of u1v1
+ u2v2 + ...
Proof
(in text) is by law of cosines
Properties
Commutative
Distributive
k(u dot v): the
scalar can multiply either vector (it doesn't distribute to both)
0
vector dot v = 0 scalar
unexpected: v dot v =
(norm of v)2
Proof
(component-wise, for R3):
Let
v = <v1,v2,v3>,
calculate LHS and RHS, equate
Angle between vectors:
cosq = dot product / product of norms
Ex:
Note
that a negative cosine gives us an obtuse angle (cosq is neg if q is in second quadrant)
Ex: Check base angle in tetrahedron, between
<2,0,0> and (<1,sqrt(3),0>
Vector projection onto
unit vector e:
magnitude is always v dot e; direction is e
arbitrary vector b:
magnitude is v dot b/norm(b); direction is b/norm(b)
Go
over notation projvb and v- projvb
Distance between a point and a line in
2-space
Cosines of direction angles (between vector
and coord axes):
Take
dot product with i, j, k
Result: cos α = v1
/ norm(v),
cos β = v2 / norm(v),
cos γ = v3 / norm(v)
3.4 Cross Product Hwk: 1-22 (part b only, except do all of #9), 25b, 29, 36-39, T1
Standard Unit Vectors i, j, k
Calculation of
2X2 determinant
3X3 determinant by expansion across the first row
Key properties (easily demonstrated for 2X2)
If two rows are equal, det is 0
Row swap changes the sign
Def of cross product in terms of a determinant
Ex: calculate a sample cross product
Thm 1 Relationship with dot product. Proofs: component-wise
a,b. Cross prod of two vectors is orthog to both vectors, with direction given by right-hand rule
c. Lagrange's identity (used to prove sine formula)
d,e. relationship between cross- and dot-products
Thm 2 Properties of cross-product
a. Anticommutative
b. L and R Distrib
c. Scalar can apply to either factor
d. Identity is 0 vector
e. Cross prod of a vector with itself is 0
Cross prods of unit vectors
Illustrate with circuit i..j..k..and back to i
Thm
Cross-prod formula for sinq Proof follows from Lagrange's identity and cosine formula
Area of parallelogram Follows from triangle trig
Cross prod is 0 for parallel vectors
Scalar triple product
Volume of parallelepiped
Test for coplanar vectors
Go to Cross_Product for an interactive internet 3D visualization of the cross product
3.5 Lines and Planes in 3-Space Hwk: 1ac, 2a, 5ab, 8, 9a, 10a, 11, 14, 17, 19, 25, 29, 34, 37, 39a, 42, 45, 47, 48
From MATH-04C: PARAMETRIC EQUATIONS OF LINES
Eqn of line through P0 (x0,y0,z0) parallel to vector v = <a,b,c>, with P (x,y,z) an arbitrary point on line.
1. Vector forms:
Concept vector P0P is parallel to v
P0P = tv for parameter t; as t ranges over real nrs, you get every point on the line
Expand: <x-x0,y-y0,z-z0> = t <a,b,c>
Solve: <x,y,z> = <x0,y0,z0> + t <a,b,c> Easiest form for me to remember
Interpret: Get to arbitrary point (x,y,z) by starting at fixed point and adding scalar mult of v.
2. Parametric eqns: Get eqn for each component in vector form:
x = x0 + at
y = y0 + bt
z = z0 + ct
3. Symmetric eqns: Solve all for t, equate: (x-x0) / a = (y-y0) / b = (z-z0) / c
Example: for some arbitrary P0 = ( ___ ) , v = < ___ >
Parallel lines: vector portions <a,b,c> are scalar multiples of one another
Intersecting lines: Simultaneous eqns, but you must make sure to use different letters (or subscripts) for the parameters for the two lines.
Ex: L1: x = 2 + t, y = 2 + 3t, z = 3 + t
L2: x = 4t + 1, y = 6t + 5, z = 2t + 4. Replace t in L2's eqns by u
Solve simult eqns for x and y, then use the values of t and u to calculate z
If both z's are the same, lines intersect.
If the z's differ, lines are skew.
Line segment joining two points:
Where does v come from?
Specify values for parameter t (0 and 1 if built in the standard manner) to designate endpoints.
Vector form of eqn in text: r = r0 + tv
Relate to form we started with
Use of r for "vector valued function"
Sample test questions:
Let u = 2i-j+k, v
= 3i+3j-k, w =
i+4j-2k. Find
The cosine of the angle
between u and v. [This checks dot products]
A unit vector in the direction of w.
The vector component of v in the direction of w (i.e.
the scalar projection of v onto w)
projwv.
Determine whether u
and w are orthogonal, parallel, or
neither.
A vector orthogonal to
both u and v. [This checks cross products]
The area of the triangle
determined by u and v.
The volume of the oblique box (parallelepiped)
determined by u, v, and w.
A unit vector in the xy-plane that is perpendicular to w.
[What vector would you
have to cross w with?]
Determine whether u,
v, and w lie in the same plane.
From MATH-04C: PLANES IN 3-SPACE
Equation of plane: Concept: normal vector and displacement vector (from fixed point (x0,y0,z0) to arbitrary point (x,y,z) ) are perpendicular:
Vector form using dot product of a normal vector and a displacement vector:
n dot (r-r0) = 0
Same thing in component form:
<a,b,c> dot (x-x0, y-y0, z-z0> = 0
Take dot product to get point-normal form:
a(x-x0) + b(y-y0) + c(z-z0) = 0
Clear paren and collect like terms to get general form:
ax + by +cz + d = 0, where a normal vector is n = <a,b,c>
Example ...
Angle between planes = angle between normals
Find cos q where q is the angle between n1 and n2
Example ...
Distance between point P (x0,y0,z0) and plane ax+by+cz+d = 0
Concept: pick any point Q (x1,y1,z1) in the plane.
The desired distance is the magnitude of the projection of the vector PQ (or QP) onto n
Vector formula: |QP dot n| / norm(n)
Scalar formula: 
Where does formula come from?
Work out vector formula:
QP dot n = <x0-x1, y0-y1, z0-z1> dot (<a,b,c>
= ax0 + by0 + cz0 - (ax1 + by1 + cz1)
= ax0 + by0 + cz0 - (-d) (note ax1+by1+cz1+d = 0 because Q is in the plane)
Example ...
Alternate applications
Distance between parallel planes
The points P and Q are any two points, P in the first plane, Q in the other.
Distance between skew lines
Pick any two points P and Q on the two lines
Find any two vectors v1 and v2 parallel to the two lines
The normal vector (to parallel planes containing the two lines) is v1 cross v2
Discovering equations of Planes in space - interactive internet applet
Return to: Merced College; Don Power Updated 02/28/07 by Don Power