p.10
Projection of Vectors
What is the formula for the scalar projection of vector A along vector u?
‖proj u A‖ = |u · A / ‖u‖|.
p.5
Scalar Multiplication of Vectors
What is the result of multiplying the vector u = 3i + 4j − k by 2/3?
2/3 u = 2i + (8/3)j − (2/3)k.
p.7
Dot Product of Vectors
How is the norm of a vector related to the dot product?
The norm is equal to the square root of the dot product of the vector with itself: ∥ ~u ∥ = √(~u · ~u).
p.6
Dot Product of Vectors
What is another name for the dot product?
It is also called the inner product or scalar product.
p.2
Vector Addition and Properties
What is the parallelogram rule in vector addition?
It states that vectors ~a and ~b form the sides of a parallelogram, and the diagonal represents the sum ~a + ~b.
p.8
Unit Vectors and Normalization
How do you find the unit vector in the direction of vector ~v = ⟨-2, 5, 3⟩?
ˆu = ~v / ||~v|| = 1/√38 ⟨-2, 5, 3⟩
p.7
Magnitude and Norm of Vectors
What is the norm of a vector?
The length of the vector, denoted by ∥ ~u ∥, defined as ∥ ~u ∥ = √(u1² + u2² + u3²).
p.3
Scalar Multiplication of Vectors
What is scalar multiplication?
It is the operation of multiplying a vector by any scalar c, resulting in c~a = 〈ca1, ca2, ca3〉.
p.6
Vector Addition and Properties
What is the theorem regarding the addition of vectors?
1. ~v + ~w = ~w + ~v (Commutative property). 2. (~v + ~w) + ~u = ~v + (~w + ~u) (Associative property). 3. ~v + ~0 = ~v (Identity element). 4. ~v + ~(-v) = ~0 (Inverse element).
p.3
Component Form of Vectors
How can a vector ~a in R3 be expressed in terms of standard unit vectors?
~a = a1e1 + a2e2 + a3e3, where e1, e2, e3 are the standard basis vectors.
p.2
Component Form of Vectors
What is the component form of a vector in three dimensions?
The component form of vector ~u is given by ~u = ⟨u1, u2, u3⟩.
p.8
Magnitude and Norm of Vectors
When is a vector ~u equal to the zero vector?
~u = 0 if and only if ||~u|| = 0
p.6
Dot Product of Vectors
What is the result of the dot product for vectors ~a = ⟨0, 3, -7⟩ and ~b = ⟨2, 3, 1⟩?
~a · ~b = 0 × 2 + 3 × 3 + (-7) × 1 = 0 + 9 - 7 = 2.
p.3
Definition of Scalars and Vectors
When are two vectors considered equal?
Two vectors are equal if they have equal magnitude and the same direction, or if their coordinates are equal.
p.6
Dot Product of Vectors
What is the definition of the dot product of two vectors?
The dot product of vectors ~a = ⟨a1, a2, a3⟩ and ~b = ⟨b1, b2, b3⟩ is defined as ~a · ~b = a1b1 + a2b2 + a3b3.
p.7
Unit Vectors and Normalization
How do you create a unit vector from a non-zero vector?
By dividing the vector by its length: ˆu = ~u / ∥ ~u ∥.
What defines two vectors as opposite?
They have opposite directions but not necessarily the same magnitude.
p.4
Vector Addition and Properties
In the example provided, are the vectors ~a = ⟨4, 10⟩ and ~b = ⟨2, -9⟩ parallel?
No, they are not parallel as they cannot be expressed as scalar multiples of each other.
p.6
Dot Product of Vectors
How do you compute the dot product of vectors ~v = 5i - 8j and ~w = i + j?
~v · ~w = 5 × 1 + (-8) × 1 = 5 - 8 = -3.
p.7
Dot Product of Vectors
What is the distributive property of the dot product?
The property states that ~u · (~v + ~w) = (~u · ~v) + (~u · ~w).
p.1
Definition of Scalars and Vectors
What defines a vector?
A vector is a physical quantity that has both magnitude and direction.
p.1
Definition of Scalars and Vectors
What do the points P and Q represent in vector notation?
Point P is the initial point (origin) and point Q is the terminal point (endpoint).
p.10
Projection of Vectors
How is the projection of vector A onto vector u defined mathematically?
proj u A = (u · A / ‖u‖²) u.
p.10
Projection of Vectors
What is the first step to calculate proj v u?
Calculate the dot product v · u.
p.7
Definition of Scalars and Vectors
What is the zero vector?
The vector with length zero, written as 〈0, 0, 0〉, denoted by 0, with an arbitrary direction.
p.9
Angle Between Two Vectors
What condition indicates that two vectors ~u and ~v are orthogonal?
The angle between them is π/2, which means ~u · ~v = 0.
How can vector subtraction be performed using addition?
By adding the opposite of the second vector to the first vector: ~a - ~b = ~a + (-1 ~b).
p.1
Component Form of Vectors
What is the component form of a vector in the plane?
If ~v has its initial point at the origin and terminal point at (v1, v2), then ~v = ⟨v1, v2⟩.
p.3
Vector Addition and Properties
What is the result of adding two vectors ~a and ~b?
The result is ~a + ~b, which can also be represented as ~b + ~a due to the commutative property.
p.10
Projection of Vectors
Given vectors u = 2i - j + 3k and v = i + 2j + k, what is proj v u?
proj v u = (v · u / ‖v‖²) v.
p.2
Component Form of Vectors
How is a vector from the origin to a point in two dimensions represented?
It is written as ~v = ⟨v1, v1⟩.
p.9
Angle Between Two Vectors
How can the cosine of the angle between two vectors be expressed?
cos θ = ~u · ~v / (||~u|| ||~v||).
How can you express vector subtraction mathematically?
The difference of vectors ~a and ~b is given by ~a - ~b = (a1 - b1)e1 + (a2 - b2)e2 + (a3 - b3)e3.
p.3
Component Form of Vectors
What are the standard basis vectors in three-dimensional space?
They are e1 = i = 〈1, 0, 0〉, e2 = j = 〈0, 1, 0〉, and e3 = k = 〈0, 0, 1〉.
p.10
Projection of Vectors
What is the significance of the angle between vectors when discussing projections?
It determines whether the projection is acute or obtuse.
p.2
Vector Addition and Properties
How is the sum of two vectors ~a and ~b defined?
The sum is given by ~a + ~b = ⟨a1 + b1, a2 + b2, a3 + b3⟩.
p.9
Angle Between Two Vectors
What can be inferred if the dot product of two non-zero vectors is zero?
The vectors are orthogonal.
p.7
Definition of Scalars and Vectors
What is a unit vector?
A vector with a length or magnitude of one, used to indicate direction, often denoted with a hat (ˆu).
p.9
Angle Between Two Vectors
What is the relationship between the angle θ and the dot product of two non-zero vectors ~u and ~v?
If θ is the angle between two non-zero vectors ~u and ~v, then ~u · ~v = ||~u|| ||~v|| cos θ.
p.4
Vector Addition and Properties
Under what condition are two vectors considered parallel?
If one vector is a scalar multiple of the other.
p.10
Projection of Vectors
What does the projection of a vector represent in geometric terms?
It represents the shadow of the vector on the line containing another vector.
p.2
Vector Addition and Properties
What graphical method is used to represent vector addition?
Place the tail of vector ~b at the head of vector ~a and draw an arrow from the tail of ~a to the head of ~b.
p.8
Magnitude and Norm of Vectors
How do you calculate the magnitude of the vector from P to Q?
||→PQ|| = √((-5)² + (12)² + (2)²) = √173
p.1
Definition of Scalars and Vectors
How is the magnitude of a vector represented in a diagram?
By the length of the arrow, which is proportional to the vector's magnitude.
p.2
Component Form of Vectors
What is the general form of a vector in n-dimensional Euclidean space?
The vector is represented as ~a = ⟨a1, a2, a3, ..., an-1, an⟩.
p.7
Dot Product of Vectors
What is the commutative property of the dot product?
The property states that ~u · ~v = ~v · ~u.
p.1
Definition of Scalars and Vectors
How are vectors typically denoted?
By lowercase boldface letters, such as a or ~a.
p.1
Component Form of Vectors
How is a vector in three-dimensional space represented?
If ~u has its initial point at the origin and terminal point at (u1, u2, u3), it is represented accordingly.
p.1
Definition of Scalars and Vectors
What are scalars?
Quantities that can be described by magnitude only, such as mass, length, and speed.
What is the geometric interpretation of vector subtraction?
It can be visualized by placing the tails of both vectors at the same point and drawing an arrow from the head of ~b to the head of ~a.
p.1
Definition of Scalars and Vectors
What is an alternative definition of a vector?
A directed line segment that has an initial and terminal point.
