Q3Comprehension4 Marks15 Mar 2026PassageConsider a subset W={(x,y,4)∈R3∣x,y∈R}W=\{(x,y,4)\in\mathbb{R}^3\mid x,y\in\mathbb{R}\}W={(x,y,4)∈R3∣x,y∈R} of R3\mathbb{R}^3R3. Based on the above data, answer the given subquestions.Which of the following options is correct?A(W,+,⋅)(W,+,\cdot)(W,+,⋅) is a vector space with addition +:W×W→W+:W\times W\to W+:W×W→W defined as (x1,y1,4)+(x2,y2,4)=(x1+x2,y1+y2,4)(x_1,y_1,4)+(x_2,y_2,4)=(x_1+x_2,y_1+y_2,4)(x1,y1,4)+(x2,y2,4)=(x1+x2,y1+y2,4) and scalar multiplication ⋅:R×W→W\cdot:\mathbb{R}\times W\to W⋅:R×W→W defined as c⋅(x,y,4)=(cx,cy,4c)c\cdot(x,y,4)=(cx,cy,4c)c⋅(x,y,4)=(cx,cy,4c).B(W,+,⋅)(W,+,\cdot)(W,+,⋅) is a vector space with addition +:W×W→W+:W\times W\to W+:W×W→W defined as (x1,y1,4)+(x2,y2,4)=(x1,y1,4)(x_1,y_1,4)+(x_2,y_2,4)=(x_1,y_1,4)(x1,y1,4)+(x2,y2,4)=(x1,y1,4) and scalar multiplication ⋅:R×W→W\cdot:\mathbb{R}\times W\to W⋅:R×W→W defined as c⋅(x,y,4)=(cx,cy,4)c\cdot(x,y,4)=(cx,cy,4)c⋅(x,y,4)=(cx,cy,4).C(W,+,⋅)(W,+,\cdot)(W,+,⋅) is a vector space with addition +:W×W→W+:W\times W\to W+:W×W→W defined as (x1,y1,4)+(x2,y2,4)=(x1+x2,y1+y2,4)(x_1,y_1,4)+(x_2,y_2,4)=(x_1+x_2,y_1+y_2,4)(x1,y1,4)+(x2,y2,4)=(x1+x2,y1+y2,4) and scalar multiplication ⋅:R×W→W\cdot:\mathbb{R}\times W\to W⋅:R×W→W defined as c⋅(x,y,4)=(cx,cy,4)c\cdot(x,y,4)=(cx,cy,4)c⋅(x,y,4)=(cx,cy,4).D(W,+,⋅)(W,+,\cdot)(W,+,⋅) is a vector space with addition +:W×W→W+:W\times W\to W+:W×W→W defined as (x1,y1,4)+(x2,y2,4)=(x1,y1,4)(x_1,y_1,4)+(x_2,y_2,4)=(x_1,y_1,4)(x1,y1,4)+(x2,y2,4)=(x1,y1,4) and scalar multiplication ⋅:R×W→W\cdot:\mathbb{R}\times W\to W⋅:R×W→W defined as c⋅(x,y,4)=(cx,cy,4c)c\cdot(x,y,4)=(cx,cy,4c)c⋅(x,y,4)=(cx,cy,4c).
