Linear Algebra Chapter 3
k(u × v) = ?
(k u) × v
k(mu) = ?
(km)u
k(u ⋅ v) = ?
(ku) ⋅ v or u ⋅ (kv)
u × (v + w) = ?
(u × v) + (u × w)
(u + v) × w = ?
(u × w) + (v × w)
Vector component of u along a (proj(suba)u)
(u ⋅ a) * a/||a||^2
u × (v × w) = ?
(u ⋅ w)v − (u ⋅ v)w
Why are collinear vectors with the same length not equal?
-v and v are not equal, but have same length
The dot product of any of two of the R^3 standard unit vectors is equal to what?
0
u × 0 = ?
0
u × u = ?
0
v ⋅ v = 0 iff v =?
0
θ is acute if u ⋅ v > ?
0
How to find vector equation given point (x0,y0,z0) and vectors <v11,v12, v13> and <v21, v22, v23>?
1) Plug into x = x0 + t1v1 + t2v2 2) <x, y, z> = <x0, y0, z0> + t1<v11,v12, v13> + t2<v21, v22, v23>
Process for finding cross product of 2 vectors in R^3
1. Create matrix 2x3 matrix 2. Block of the col corresponding the component you're on (first comp - block out col 1) 3. Find the det of the resulting 2x2 4. If component 2, make det negative
Process for finding area of triangle given three points using cross product
1. Create vectors from points: P1P2 and P1P3 2. Perform cross product P1P2 x P1P3 3. Find norm of P1P2 x P1P3 4. Multiply norm of P1P2 x P1P3 by 1/2
Process for finding cross product with i, j, k?
1. Take det of 3x3 with i, j, k in first row 2. Expand on row 1 3. Combine the result to vector form or leave in i-j+k form
Dot product in terms of magnitude of the two vectors
1/4(‖u + v‖^2) - 1/4‖u - v‖^2
‖u + v‖^2 + ‖u - v‖^2 = ?
2(‖u‖^2 + ‖v‖^2)
How you find the components of a vector given a set of points?
<tip x1 - tail x1, tip x2 - tail x2, ... , tip xn - tail xn>
u x v =
<u2v3-u3v2, u3v1 - u1v3, u1v2 - u2v1>
What is a unit vector?
A vector with a magnitude of 1
Area of a parallelogram
A= (b) (h) = (‖u‖) (‖v‖ sinθ)
i x j, j x k, k x i equal what? What about when flipped?
Equal the standard vector not included (i x j = k), if you flip the order (j x i) equals the negative of the vector
If there are two points x0 and x1, how do you find the vector parallel to them?
Find vector with the points by subtracting, parallel vectors can be the same, v = x1-x0
How to find cross product with vectors in R2?
Make the z coordinate zero for both vectors
Why can the vector equation of a plane not be determined from any point lying in the plane and a nonzero vector parallel to the plane?
Need two vectors
If three vectors have the same initial point, then how do you know that are in the same plane or not?
Scalar triple product equals 0
Direction of vector arrow is what?
The direction of the vector
The tail is what part of the vector?
The initial point
The length of a vector arrow is what?
The magnitude of the vector
How can we translate the matrix notation of equations (Ax=b) to vectors?
The notation is the same as doing the dot product of coefficient vector a with x, a ⋅ x = b
If Ax=0, then what is true for every row vector in A?
They are orthogonal to every vector Ax=0
Two planes are parallel if what is true about their normal vectors?
They are scalar multiples of each other
Point normal equation of a plane with normal vector <a,b,c>
a(x-x0) + b(y-y0) + c(z-z0) = 0
Point normal equation of a line with normal vector <a,b>
a(x-x0) + b(y-y0) = 0
What is the volume of a parallelepiped?
abs(scalar triple product)
What is the absolute value of a the determinant of a 2x2 matrix made with each row containing the components of a vector?
area of a parallelogram
The scalar triple product can be expressed as what?
determinant of 3x3
Equivalent vectors can have a different amount of components
false
If u, v, and w are vectors in R3, where u is nonzero and u × v = u × w, then v = w.
false
In general, u × (v × w) = (u × v) × w
false
The cross product is commutative (u x v = v x u)
false
The norm of a vector can be negative
false
The orthogonal projection of u on a is the same as the vector component of u orthogonal to a
false
If u ⋅ v < 0, then what is true of θ?
it is obtuse
(k+m)u = ?
ku + mu
The dot product results in what?
number
u ⋅ (v × w) results in what type of output?
scalar
In x-x0 = tv, what is t?
scalar, called parameter
Norm of vector
sqrt(sum of squared components)
i hat, j hat, and k hat are what?
standard unit vectors for R^3
In vector = AB, what is A?
tail
The tip is what part of the vector?
the arrowhead/terminal point
||u - v|| = ?
the distance between the components of u and v
The norm of a vector is the same as what?
the magnitude or length
If u ⋅ v = 0 and u and v are not equal to zero, then what is true of u and v?
they are orthogonal
In vector = AB, what is B?
tip
A normal vector to a plane can be obtained by taking the cross product of two nonzero and noncollinear vectors lying in the plane.
true
All R^3 standard unit vectors are orthogonal to one another
true
All solution vectors of the linear system Ax = b are orthogonal to the row vectors of the matrix A if and only if b = 0.
true
Equivalent vectors have the same direction
true
If two vectors are equivalent, then they are the same direction and length
true
The norm of a vector is zero iff that vector is the zero vector
true
The orthogonal projection of u on a is the same as the vector component of u along a
true
The points lying on a line through the origin in R2 or R3 are all scalar multiples of any nonzero vector on the line.
true
The scalar triple can be done in any order of three vectors and get the same result
true
The zero vector has no natural direction, so it can be assigned any direction
true
The zero vector is orthogonal to all vectors
true
v + w = w + v
true
Vector component of u orthogonal to a (u - proj(suba)u)
u - ((u ⋅ a) * a/||a||^2)
If u ⋅ (u x v) = 0, what is the relationship between u x v and u?
u x v is orthogonal to u
If v ⋅ (u x v) = 0, what is the relationship between u x v and v?
u x v is orthogonal to v
(k u) × v = ?
u × (k v)
− (v × u) = ?
u × v
Scalar Triple Product
u ⋅ (v × w)
u ⋅ (v + w) = ?
u ⋅ v + u ⋅ w
cosθ = ?
u ⋅ v/||u|| ||v||
Dot product using components
u1v1+u2v2+...+unvn
How to find a line passing through two points x0 and x1 and parallel to vector v?
v = x1-x0, x = x0+t(x1-x0) = (1-t)x0 + tx1
u ⋅ v = ?
v ⋅ u
How do you find the unit vector for v?
v/||v||
The cross product results in what?
vector
What is the absolute value of a the determinant of a 3x3 matrix made with each row containing the components of a vector?
volume of a parallelepiped
When are vectors collinear?
when they are parallel or on the same line
How to find plane that goes through point x0 and is parallel to a vectors v1 and v2?
x = x0 + t1v1 + t2v2
How to find line that goes through point x0 and is parallel to a vector v?
x = x0 + tv
How can you find the line segment from vectors x0 to x1 when 0 ≤ t ≤ 1?
x = x0+t(x1-x0) = (1-t)x0 + tx1
Is the zero vector parallel to any vector?
yes
||kv|| = ?
|k| ||v||
Length of orthogonal projection of u onto a
||proj(suba)u|| = |u ⋅ a|/||a||
Length of orthogonal projection of u onto a involving angles
||u|| ||cosθ||
Dot product using angles
||u|| ||v|| cosθ
If u and v are orthogonal, then ||u+v||^2 = ?
||u||^2 + ||v||^2
If u ⋅ v = 0, then θ = ?
π/2
‖u + v‖ ≤ ?
‖u‖ + ‖v‖
According to Cauchy-Schwarz inequality, |u ⋅ v| ≤ ?
‖u‖ ‖v‖
‖u × v‖ = ?
‖u‖ ‖v‖ sinθ
‖u × v‖^2 = ?
‖u‖^2‖v‖^2 − (u ⋅ v)^2
Distance between two points
√(x₁-x₂)² + (y₁-y₂)²
