Multivariable Calculus: Vectors and 3D Space
What is the sum of two vectors called?
the resultant
Given *v*=*i*-2*j*+2*k* and *u*=4*i*-3*k* find the projection of *v* in the direction of *u*.
(-8/25)*i*+(6/25)*j*
What is the intersection of the line P(t)=(2t-1, 3, -t+7) with the plane 3x-2y+z=10?
(19/5, 3, 23/5)
How are points expressed in 3D space?
(x, y, z)
What is the equation of a sphere centered at (3, 2, -4) with r=2?
(x-3)²+(y-2)²+(z+4)²=4
How are vectors calculated in 3D space given two points, where P=(x₁, y₁, z₁) and Q=(x₂, y₂, z₂)?
*PQ*=⟨x₂-x₁, y₂-y₁, z₂-z₁⟩
What are the vector properties?
*a*+*b*=*b*+*a* *a*+0=*a* c(*a*+*b*)=c*a*+c*b* cd(*a*)=c(d*a*) (*a*+*b*)+*c*=*a*+(*b*+*c*) *a*+-*a*=0 1*a*=*a*
What are the dot product properties?
*a*·*a*=||*a*||² *a*·*b*=*b*·*a* *a*·(*b*+*c*)=*a*·*b*+*a*·*c* (c*a*)·*b*=c(*a*·*b*)=*a*·(c*b*) 0·*a*=0
What are the properties of cross products?
*u*×*v*=-(*v*×*u*) (*a*+*b*)×*w*=*a*×*w*+*b*×*w*
How is the unit vector *v* calculated?
*v*/||*v*||
What are the two methods of notation for vectors?
*v*=⟨a, b, c⟩=a*i*+b*j*+c*k*
If *v*=⟨v₁, v₂, v₃⟩ and *u*=⟨u₁, u₂, u₃⟩, what is *v*·*u*?
*v*·*u*=v₁u₁+v₂u₂+v₃u₃
Given *v*=*i*-2*j*+2*k* and *u*=4*i*-3*k* find the component of *u* in the direction of *v*.
-2/3
What is the equation of the plane defined by P(3, 2, 1), Q(5, 0, -1), and R(4, 3, 2)?
-y+z=-1
What is (2*i*-3*j*+4*k*)×(*i*+*j*-7*k*)?
17*i*+18*j*+5*k*
What is ⟨1, 2, 3⟩·⟨4, 5, 6⟩
32
What do two points in space create?
A vector or an axis in which a plane could be drawn through
What is the result of a cross product?
A vector that is orthogonal to each vector
What is the formula for the area of a parallelogram spanned by *u* and *v*?
A=||*u*x*v*||
What is the formula for the area of a triangle spanned by *u* and *v*?
A=½||*u*x*v*||
What are the eight regions divided by the three planes of 3D space called?
Octants
What is the projection of vectors?
The "mapping" of on vector onto another by splitting it into it's components based on the other vector
If a dot product is 0, what is true about the vectors?
They are orthogonal
If *u*×*v*=0, what is true about *u* and *v*?
They are parallel
How are vectors added (visually)?
Tip-to-tail
What is the parametrization of a line from (0, 2, -3) to (7, 5, 1)?
c(t)=(7t, 3t+2, 4t-3), t∈[0, 1]
What is the equation for the magnitude of projᵥ*u*?
compᵥ*u*=||*u*||cosθ=(*u*·*v*)/||*v*||
What is the formula for the angle between two vectors?
cosθ=(*u*·*v*)/(||*u*||·||*v*||)
How are vectors added (mathematically)?
if *u*=⟨u₁, u₂⟩ and *v*=⟨v₁, v₂⟩, then *u*+*v*=⟨u₁+v₁, u₂+v₂⟩
What is the equation for projᵥ*u*?
projᵥ*u*=[(*u*·*v*)/||*v*||²]·*v*
What is the magnitude of *PQ*, where Q=(x₁, y₁, z₁) and P=(x₂, y₂, z₂)?
||*PQ*||=√[(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²]
What is the area of the parallelogram spanned by *u*=⟨4, 7, 2⟩ and *v*=⟨2, 3, 1⟩?
√5
What is the area of the triangle spanned by *u*=⟨4, 7, 2⟩ and *v*=⟨2, 3, 1⟩?
√5/2
Find the normal vector of the plane defined by P(3, 2, 1), Q(5, 0, -1), and R(4, 3, 2).
⟨0, -1, 1⟩