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