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Questions and Answers
What is the result of subtracting vector b from vector a?
What is the result of subtracting vector b from vector a?
- a - b = (a1 - b1, a2 - b2) (correct)
- a - b = (a1 * b1, a2 * b2)
- a - b = (a1 + b1, a2 + b2)
- a - b = (-a1 + b1, -a2 + b2)
What is the commutative property of scalar multiplication?
What is the commutative property of scalar multiplication?
- k(a + b) = ka + kb
- kl(a) = k(la)
- (a + b)k = ak + bk
- k(a) = ak (correct)
What is the geometric interpretation of the dot product?
What is the geometric interpretation of the dot product?
- a · b = |a| + |b|
- a · b = |a||b|sin(θ)
- a · b = |a||b|cos(θ) (correct)
- a · b = |a| - |b|
What is the result of adding vector a and vector b?
What is the result of adding vector a and vector b?
What is the distributive property of scalar multiplication?
What is the distributive property of scalar multiplication?
What is the property of the dot product that states a · b = b · a?
What is the property of the dot product that states a · b = b · a?
What is the result of multiplying a vector by a scalar?
What is the result of multiplying a vector by a scalar?
What is the purpose of the dot product?
What is the purpose of the dot product?
What is the associative property of scalar multiplication?
What is the associative property of scalar multiplication?
What is the result of subtracting vector b from vector a?
What is the result of subtracting vector b from vector a?
What is the positive definiteness property of the dot product?
What is the positive definiteness property of the dot product?
What is the distributive property of the dot product?
What is the distributive property of the dot product?
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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:
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