Vectors & Space Geometry (Part 1) Flashcards
How many octants does the space have?
It has 8
How can you get the distance between two points?
By using Pythagoras. That is, using the square root of the squares of the difference between the coordinates of the points
What is the equation of a sphere?
Like the one for the circumference but with an extra term that depends on z
How can we describe the coordinates planes XY, YZ, XZ?
Z =0
X = 0
Y = 0
What is the norm of a vector?
It is the distance from the origin to the point.
How can we get the angle of a vector?
By using arctangent of y divided by x.
It can also be found by solving for theta both for the dot product or cross product
What is the dot product?
It’s the sum of the product between the respective coordinates of the involved vectors
It’s the product of the norms of the vectors times the cosine of the angle between them
What is the cross product?
It’s another vector that is orthogonal to the vectors involved in the product
It’s the product of the norms of the vectors times the sine of the angle between them
Cauchy-Schwartz and Triangular Inequality
|a*b| <= ||a|| ||b||
||a+b|| <= ||a|| + ||b||
Projection of a vector onto the other?
(a * b)/norm(a) Times a/norm(a)
What happens if we do the cross product between two vectors that are collinear?
It will result in the zero vector.
Equations of a line?
(x,y,z) = (xo, yo, zo) + t(a,b,c)
Parametric one from the one before
Symmetric one from parametric one
When are two lines parallel, orthogonal or crossed?
They are either parallel or perpendicular if their respective director vectors are either parallel or perpendicular
The cross if they’re not parallel and do not intersect
Equations for a plane
We need a point where it passes through and an orthogonal vector to the plane
n(p -po) = 0
a(x - xo) + b(y - yo) + c(z - zo) = 0
ax + by + cz - d = 0
When are two planes parallel or orthogonal?
When their respective normal vectors are parallel or orthogonal.
When two planes cross, what happens?
It generates a line that can be found by providing a value to one of the coordinates and solving for theta both remaining two.
Distance from a point to a plane?
|axo + byo + czo- d| divided by norm of normal vector
Distance from a point P1 to a line L with director vector v that passes through P0?
||(P1P0) x V|| divided ||V||
How to determine if a line intersects a plane?
If a line intersects a plane, then there must be a parameter t_{0} so that when substituted in the coordinates of L it also satisfies the equation of the plane
