Question 3 - Week 3 Practice | Prasnya
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Qualifier / Quiz 1
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Week 3
Question 3
00:00
Est. 3 min
Marks:
+4.00
0.00
Passage
Consider a subset
W
=
{
(
x
,
y
,
4
)
∈
R
3
∣
x
,
y
∈
R
}
W=\{(x,y,4)\in\mathbb{R}^3\mid x,y\in\mathbb{R}\}
of
R
3
\mathbb{R}^3
. Based on the above data, answer the given subquestions.
Question
Which of the following options is correct?
Comprehension
Medium Difficulty
15 March 2026
A
(
W
,
+
,
⋅
)
(W,+,\cdot)
is a vector space with addition
+
:
W
×
W
→
W
+:W\times W\to W
defined as
(
x
1
,
y
1
,
4
)
+
(
x
2
,
y
2
,
4
)
=
(
x
1
+
x
2
,
y
1
+
y
2
,
4
)
(x_1,y_1,4)+(x_2,y_2,4)=(x_1+x_2,y_1+y_2,4)
and scalar multiplication
⋅
:
R
×
W
→
W
\cdot:\mathbb{R}\times W\to W
defined as
c
⋅
(
x
,
y
,
4
)
=
(
c
x
,
c
y
,
4
c
)
c\cdot(x,y,4)=(cx,cy,4c)
.
B
(
W
,
+
,
⋅
)
(W,+,\cdot)
is a vector space with addition
+
:
W
×
W
→
W
+:W\times W\to W
defined as
(
x
1
,
y
1
,
4
)
+
(
x
2
,
y
2
,
4
)
=
(
x
1
,
y
1
,
4
)
(x_1,y_1,4)+(x_2,y_2,4)=(x_1,y_1,4)
and scalar multiplication
⋅
:
R
×
W
→
W
\cdot:\mathbb{R}\times W\to W
defined as
c
⋅
(
x
,
y
,
4
)
=
(
c
x
,
c
y
,
4
)
c\cdot(x,y,4)=(cx,cy,4)
.
C
(
W
,
+
,
⋅
)
(W,+,\cdot)
is a vector space with addition
+
:
W
×
W
→
W
+:W\times W\to W
defined as
(
x
1
,
y
1
,
4
)
+
(
x
2
,
y
2
,
4
)
=
(
x
1
+
x
2
,
y
1
+
y
2
,
4
)
(x_1,y_1,4)+(x_2,y_2,4)=(x_1+x_2,y_1+y_2,4)
and scalar multiplication
⋅
:
R
×
W
→
W
\cdot:\mathbb{R}\times W\to W
defined as
c
⋅
(
x
,
y
,
4
)
=
(
c
x
,
c
y
,
4
)
c\cdot(x,y,4)=(cx,cy,4)
.
D
(
W
,
+
,
⋅
)
(W,+,\cdot)
is a vector space with addition
+
:
W
×
W
→
W
+:W\times W\to W
defined as
(
x
1
,
y
1
,
4
)
+
(
x
2
,
y
2
,
4
)
=
(
x
1
,
y
1
,
4
)
(x_1,y_1,4)+(x_2,y_2,4)=(x_1,y_1,4)
and scalar multiplication
⋅
:
R
×
W
→
W
\cdot:\mathbb{R}\times W\to W
defined as
c
⋅
(
x
,
y
,
4
)
=
(
c
x
,
c
y
,
4
c
)
c\cdot(x,y,4)=(cx,cy,4c)
.
