12 Questions
What is the result of subtracting vector b from vector a?
a - b = (a1 - b1, a2 - b2)
What is the commutative property of scalar multiplication?
k(a) = ak
What is the geometric interpretation of the dot product?
a · b = |a||b|cos(θ)
What is the result of adding vector a and vector b?
a + b = (a1 + b1, a2 + b2)
What is the distributive property of scalar multiplication?
k(a + b) = ka + kb
What is the property of the dot product that states a · b = b · a?
Commutative property
What is the result of multiplying a vector by a scalar?
A new vector with the same direction
What is the purpose of the dot product?
To measure the similarity between two vectors
What is the associative property of scalar multiplication?
(k * l) * a = k * (l * a)
What is the result of subtracting vector b from vector a?
a - b
What is the positive definiteness property of the dot product?
a · a ≥ 0 and a · a = 0 if a = 0
What is the distributive property of the dot product?
a · (b + c) = a · b + a · c
Study Notes
Vector Operations
-
Addition: Two or more vectors can be added by adding corresponding components.
- Example:
a = <a1, a2>andb = <b1, b2>thena + b = <a1 + b1, a2 + b2>
- Example:
-
Subtraction: One vector can be subtracted from another by subtracting corresponding components.
- Example:
a = <a1, a2>andb = <b1, b2>thena - b = <a1 - b1, a2 - b2>
- Example:
-
Equality: Two vectors are equal if and only if their corresponding components are equal.
- Example:
a = <a1, a2>andb = <b1, b2>thena = bif and only ifa1 = b1anda2 = b2
- Example:
Scalar Multiplication
-
Scalar multiplication: A vector can be multiplied by a scalar (number) to change its magnitude and/or direction.
- Example:
a = <a1, a2>andkis a scalar, thenka = <ka1, ka2>
- Example:
-
Properties:
- Distributive property:
k(a + b) = ka + kb - Associative property:
(kl)a = k(la) - Commutative property:
k(a + b) = (ka) + (kb)
- Distributive property:
Dot Product
-
Dot product (Scalar Product): The dot product of two vectors is a scalar value that can be used to find the angle between two vectors.
- Example:
a = <a1, a2>andb = <b1, b2>thena · b = a1b1 + a2b2
- Example:
-
Properties:
- Commutative property:
a · b = b · a - Distributive property:
a · (b + c) = a · b + a · c -
Geometric interpretation: The dot product of two vectors can be used to find the angle between them:
a · b = |a||b|cos(θ)
- Commutative property:
Test your understanding of vector operations, including addition, subtraction, equality, scalar multiplication, and dot product. Learn about the properties and geometric interpretations of these operations.
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