How is a unit vector defined?
v hat = v / |v|
Geometric dot product formula?
|a||b|cos(θ)
How to know if two vectors are parallel. perp, or the angle they make?
If scalar multiples=parallel. If dot product=0, perp. If dot product=(+), acute. If dot product=(-), obtuse
Proj a onto b?
(a * b)/|b|^2, all times b vector
Area of a parallelogram?
|a x b|
Area of a triangle?
1/2 |a x b|
Volume of parallelepiped?
|a*(b x c)|
Equation of a line?
r=r0+tv
Equation of a plane?
a(x-x0)+b(y-y0)+c(z-z0)=0
How to make a plane from three points?
- Make two vectors 2. Cross them. 3. Use the plane equation
How to determine if two lines are parallel, intersect, skew, or perp?
If scalar multiple=parallel, solve system of parametric equations and if there’s a consistent solution they intersect, if not those two then they are skew, and if dot prod=0 they are perp
Equation of a sphere?
x^2+y^2+z^2=r^2
Equation of an ellipsoid?
(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1
Equation of a hyperboloid?
Theres one negative square
Equation of a paraboloid?
x^2+y^2=z
Equation of a cylinder?
Will be missing a variable
What is a trace?
Set one variable constant, then describe the shape
How do you differentiate and integrate a vector?
By components
Angle between vectors?
cos(θ)=(a * b)/|a||b|
