M1025 WEEKS 1-3
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41 Terms
Algebraic properties of vectors
What are these vectors called?
the standard basis vectors
is the dot product a number or vector?
a number
the length of a vector V is…
algebraic properties of the norm
what is normalization?
The process of finding a unit vector (Length equal to 1) in the same direction of a given vector
formula for normalization of a vector v
what do two vertical lines on either side of a vector indicate?
the normal (or square dot product of the vector)
Cauchy-schwarz inequality
Formula for the distance between two vectors
what is this equal to?
dot product of vectors u and v
what is this equal to?
=cos0
vectors u and v are ORTHOGONAL to each other if their dot product equals…
0
formula for the projection of v onto w?
formula for the projection of v onto w
what is this formula?
the equation of a line in the Cartesian plane
if a line is represented by the equation ax + by = c, what is its slope?
the coordinates of n(the normal vector) are given by what?
[a b]
![<p>[a b]</p>](https://assets.knowt.com/user-attachments/523e898f-8d98-4751-94a7-5392bfab9dd9.png)
normal form of the equation of line l
The normal form for an equation of a line l in R²
vector form of a line l in R^n
what is vector d?
direction vector of the line
what are points P and O?
P = any point on the line
O = origin
parametric eqns of lines in Rn
what is this formula? what do the variables mean?
normal form of the eqn of a plane in R³
P is any point on the plane
n is the normal vector (perpendicular to the plane)
what is this formula? what do the variables mean?
normal form of the eqn of a plane in R³
d=dot product of n & line OP
n is the normal vector (perpendicular to the plane)
[a b c]
![<p>normal form of the eqn of a plane in R³</p><ul><li><p>d=dot product of n & line OP</p></li><li><p>n is the normal vector (perpendicular to the plane) </p><ul><li><p>[a b c]</p></li></ul></li><li><p></p></li></ul><p></p>](https://assets.knowt.com/user-attachments/83f0cfd8-6f32-4ec7-8eca-579da8d698fc.png)
Two parallel planes will have
normal vectors that are multiples of one another
to build a plane, what is needed?
Two direction vectors
what is this formula? what do the variables mean?
P: any point on the plane
u & v: direction vectors that are both parallel to p
s & t: arbitrary parameters
to convert vector form to normal form in R³…
Use cross product
Shortcut for cross product?
formula for…
cross product of u and v
area of a parallelogram created by vectors u and v
|| u x v ||
area of a triangle created by vectors u and v
(1/2)||u x v||
dot product
distance from a point to a plane
