Rather than use the standard definitions of addition and scalar multiplication in R³, suppose these two operations are defined as follows. With these new definitions, is R³ a vector space? Justify your answers. (a) (X₁, X₁, Z₁) + (x2, Y2, 2₂) = (X₁ + X2, Y₁ + Y2, Z1 + Z₂) c(x, y, z)= (0, cy, cz) O The set is a vector space. O The set is not a vector space because the associative property of addition is not satisfied. vector space because it is not closed under scalar multiplication. O The set is not a O The set is not a O The set is not a vector space because the associative property of multiplication is not satisfied. vector space because the multiplicative identity property is not satisfied. (b) (x1, ₁, 2₁) + (x2, Y2, 2₂) = (0, 0, 0) c(x, y, z) = (cx, cy, cz) O The set is a vector space. O The set is not a vector space because the commutative property of addition is not satisfied. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the multiplicative identity property is not satisfied. (c) (X₁, ₁, 2₁) + (x2, Y2, 2₂) = (x₁ + x₂ + 9, Y₁ + y₂ + 9, Z₁ + Z₂ + 9) c(x, y, z) = (cx, cy, cz) O The set is a vector space. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because the additive inverse property is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the distributive property is not satisfied. (d) (x₁, Y₁, 2₁) + (x₂, Y/2, 2₂) = (X₁ + X₂ + 8, Y₁ + y₂ + 8, Z₁ + Z₂ + 8) c(x, y, z) = (cx + 8c-8, cy + 8c-8, cz + 8c-8) O The set is a vector space. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the distributive property is not satisfied. O The set is not a vector space because the multiplicative identity property is not satisfied.

Rather than use the standard definitions of addition and scalar multiplication in R³, suppose these two operations are defined as follows. With these new definitions, is R³ a vector space? Justify your answers. (a) (X₁, X₁, Z₁) + (x2, Y2, 2₂) = (X₁ + X2, Y₁ + Y2, Z1 + Z₂) c(x, y, z)= (0, cy, cz) O The set is a vector space. O The set is not a vector space because the associative property of addition is not satisfied. vector space because it is not closed under scalar multiplication. O The set is not a O The set is not a O The set is not a vector space because the associative property of multiplication is not satisfied. vector space because the multiplicative identity property is not satisfied. (b) (x1, ₁, 2₁) + (x2, Y2, 2₂) = (0, 0, 0) c(x, y, z) = (cx, cy, cz) O The set is a vector space. O The set is not a vector space because the commutative property of addition is not satisfied. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the multiplicative identity property is not satisfied. (c) (X₁, ₁, 2₁) + (x2, Y2, 2₂) = (x₁ + x₂ + 9, Y₁ + y₂ + 9, Z₁ + Z₂ + 9) c(x, y, z) = (cx, cy, cz) O The set is a vector space. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because the additive inverse property is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the distributive property is not satisfied. (d) (x₁, Y₁, 2₁) + (x₂, Y/2, 2₂) = (X₁ + X₂ + 8, Y₁ + y₂ + 8, Z₁ + Z₂ + 8) c(x, y, z) = (cx + 8c-8, cy + 8c-8, cz + 8c-8) O The set is a vector space. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the distributive property is not satisfied. O The set is not a vector space because the multiplicative identity property is not satisfied.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.4: The Dot Product

Problem 45E

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