12 Questions
What is the result of adding two or more vectors?
A new vector
What is the magnitude of a unit vector?
1
What is the result of the cross product of two vectors?
A new vector perpendicular to both
What is the property of the dot product that allows us to calculate the cosine of the angle between two vectors?
Cosine formula
What is the result of multiplying a vector by a scalar?
A scaled vector
What is the property of vector addition that allows us to rearrange the order of the vectors being added?
Commutative property
What is the geometric interpretation of the dot product of two vectors a and b?
a · b = ||a|| ||b|| cos(θ)
If a and b are two vectors, what is the resultant vector when a is subtracted from b?
b - a = b + (-a)
What is the distributive property of the cross product, and how does it relate to the vector triple product?
a × (b + c) = a × b + a × c
If a is a vector, what is the unit vector in the direction of a, and how is it related to the magnitude of a?
â = a / ||a||
What is the relationship between the scalar multiplication of a vector a by a scalar k and the magnitude of a?
The magnitude of ka is |k| times the magnitude of a
If a and b are two vectors, what is the relationship between the cross product a × b and the dot product a · b?
The cross product is anticommutative, while the dot product is commutative
Study Notes
Vector Addition
- Definition: The sum of two or more vectors, resulting in a new vector.
-
Notation:
A + BorA ⊞ B -
Properties:
- Commutative:
A + B = B + A - Associative:
(A + B) + C = A + (B + C) - Distributive:
k(A + B) = kA + kB(scalar multiplication)
- Commutative:
- Graphical Representation: Vectors are added head-to-tail, with the resulting vector being the diagonal of the parallelogram formed by the two vectors.
Magnitude and Direction
-
Magnitude (Length): The size or length of a vector, denoted as
|A|orA - Direction: The direction in which the vector points, described by an angle or unit vector
-
Unit Vector: A vector with a magnitude of 1, denoted as
û(e.g.,û = A / |A|)
Cross Product (Vector Product)
- Definition: The cross product of two vectors, resulting in a new vector perpendicular to both.
-
Notation:
A × BorA ^ B -
Properties:
- Anticommutative:
A × B = -B × A - Distributive:
A × (B + C) = A × B + A × C - Orthogonality:
A × Bis perpendicular to bothAandB
- Anticommutative:
Dot Product (Scalar Product)
- Definition: The dot product of two vectors, resulting in a scalar value.
-
Notation:
A · BorA ⋅ B -
Properties:
- Commutative:
A · B = B · A - Distributive:
A · (B + C) = A · B + A · C - Cosine Formula:
A · B = |A| |B| cos(θ)
- Commutative:
Vector Multiplication
- Scalar Multiplication: Multiplying a vector by a scalar, resulting in a scaled vector.
- Vector Multiplication: The cross product or dot product of two vectors.
Reciprocals
-
Definition: The reciprocal of a vector, denoted as
1/AorA⁻¹ -
Properties:
-
A⁻¹ A = 1(multiplicative identity) -
(kA)⁻¹ = k⁻¹ A⁻¹(scalar multiplication)
-
Projector
- Definition: A linear transformation that projects a vector onto a subspace.
-
Notation:
P_A(projector onto the subspace spanned byA)
Law
- Law of Cosines: Relates the lengths and dot product of two vectors and their sum.
- Law of Sines: Relates the lengths and direction of two vectors and their sum.
Test your understanding of vector addition, scalar multiplication, dot product, cross product, and other essential concepts in linear algebra and vector calculus.
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