What is the formula for the scalar projection of vector A along vector u?
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‖proj u A‖ = |u · A / ‖u‖|.
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What is the formula for the scalar projection of vector A along vector u?
‖proj u A‖ = |u · A / ‖u‖|.
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.
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).
What is another name for the dot product?
It is also called the inner product or scalar product.
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.
How do you find the unit vector in the direction of vector ~v = ⟨-2, 5, 3⟩?
ˆu = ~v / ||~v|| = 1/√38 ⟨-2, 5, 3⟩
What is the result of adding the vectors u = i + 4j − 2k and v = 2i − 5j?
u + v = 3i − j − k.
What is the norm of a vector?
The length of the vector, denoted by ∥ ~u ∥, defined as ∥ ~u ∥ = √(u1² + u2² + u3²).
What is scalar multiplication?
It is the operation of multiplying a vector by any scalar c, resulting in c~a = 〈ca1, ca2, ca3〉.
What is the theorem regarding the addition 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.
What is the component form of a vector in three dimensions?
The component form of vector ~u is given by ~u = ⟨u1, u2, u3⟩.
When is a vector ~u equal to the zero vector?
~u = 0 if and only if ||~u|| = 0
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.
What is the result of adding the vectors u = ⟨−1, 2, 1⟩ and v = ⟨2, −5, 0⟩?
u + v = ⟨1, −3, 1⟩.
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.
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.
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.
What is the scalar product of vectors ~u = ⟨2, 5, -1⟩ and ~v = ⟨-1, 2, 3⟩?
~u · ~v = 5
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.
What is the projection of vector A onto vector u denoted as?
proj u A.
What is the result of negating the vector u = 3i + 4j − k?
−u = −3i − 4j + k.
What is the relationship between the dot product of a vector and its magnitude?
~u · ~u = ||~u||²
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.
What is the distributive property of the dot product?
The property states that ~u · (~v + ~w) = (~u · ~v) + (~u · ~w).
What defines a vector?
A vector is a physical quantity that has both magnitude and direction.
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).
How is the projection of vector A onto vector u defined mathematically?
proj u A = (u · A / ‖u‖²) u.
What is the first step to calculate proj v u?
Calculate the dot product v · u.
What is the zero vector?
The vector with length zero, written as 〈0, 0, 0〉, denoted by 0, with an arbitrary direction.
How do you find the component form of the vector from point P to point Q?
→PQ = Q - P
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).
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⟩.
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.
Given vectors u = 2i - j + 3k and v = i + 2j + k, what is proj v u?
proj v u = (v · u / ‖v‖²) v.
How is a vector from the origin to a point in two dimensions represented?
It is written as ~v = ⟨v1, v1⟩.
What is the formula for the magnitude of a scalar multiplied by a vector?
||α~u|| = |α| ||~u||
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.
In the example provided, are the vectors ~a = ⟨2, -4, 1⟩ and ~b = ⟨-6, 12, -3⟩ parallel?
Yes, because ~b = -3 ~a.
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〉.
What is the significance of the angle between vectors when discussing projections?
It determines whether the projection is acute or obtuse.
How is the sum of two vectors ~a and ~b defined?
The sum is given by ~a + ~b = ⟨a1 + b1, a2 + b2, a3 + b3⟩.
What is the component form of the vector from P = (3, -7, 1) to Q = (-2, 5, 3)?
→PQ = ⟨-5, 12, 2⟩
What can be inferred if the dot product of two non-zero vectors is zero?
The vectors are orthogonal.
What is the condition for the angle θ if it is a right angle?
Then ~u · ~v = 0.
What is the scalar multiplication property of vectors?
(αβ) ~v = α (β ~v).
What is a unit vector?
A vector with a length or magnitude of one, used to indicate direction, often denoted with a hat (ˆu).
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.
Under what condition are two vectors considered parallel?
If one vector is a scalar multiple of the other.
What does it mean if the angle θ between two vectors is obtuse?
Then ~u · ~v < 0.
What does the projection of a vector represent in geometric terms?
It represents the shadow of the vector on the line containing another vector.
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.
How do you calculate the magnitude of the vector from P to Q?
||→PQ|| = √((-5)² + (12)² + (2)²) = √173
What does it mean if the angle θ between two vectors is acute?
Then ~u · ~v > 0.
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.
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⟩.
What is the commutative property of the dot product?
The property states that ~u · ~v = ~v · ~u.
What does the zero vector dot with any vector equal?
0 · ~u = 0.
How are vectors typically denoted?
By lowercase boldface letters, such as a or ~a.
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.
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.
What is an alternative definition of a vector?
A directed line segment that has an initial and terminal point.
