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if a 2 and b 3 and the angle between a and b is 12
Question:
If
∣
a
⃗
∣
=
2
|\vec{a}|=2
∣
a
∣
=
2
and
∣
b
⃗
∣
=
3
|\vec{b}|=3
∣
b
∣
=
3
and the angle between
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
is 120°, then the length of the vector
∣
1
a
⃗
2
−
1
b
⃗
3
∣
2
|\frac{1\vec{a}}{2}-\frac{1\vec{b}}{3}|^2
∣
2
1
a
−
3
1
b
∣
2
is
KCET - 2022
KCET
Updated On:
Dec 8, 2024
2
1
6
\frac{1}{6}
6
1
3
1
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The Correct Option is
A
Solution and Explanation
The correct answer is (A) : 2.
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2
Top Questions on Vectors
Let
a
⃗
=
i
^
+
2
j
^
+
3
k
^
,
b
⃗
=
3
i
^
+
j
^
−
k
^
\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \, \vec{b} = 3\hat{i} + \hat{j} - \hat{k}
a
=
i
^
+
2
j
^
+
3
k
^
,
b
=
3
i
^
+
j
^
−
k
^
and
c
⃗
\vec{c}
c
be three vectors such that
c
⃗
\vec{c}
c
is coplanar with
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
. If the vector
c
⃗
\vec{c}
c
is perpendicular to
b
⃗
\vec{b}
b
and
a
⃗
⋅
c
⃗
=
5
\vec{a} \cdot \vec{c} = 5
a
⋅
c
=
5
, then
∣
c
⃗
∣
|\vec{c}|
∣
c
∣
is equal to:
JEE Main - 2025
Mathematics
Vectors
View Solution
Let
a
⃗
=
i
+
j
+
k
\vec{a} = i + j + k
a
=
i
+
j
+
k
,
b
⃗
=
2
i
+
2
j
+
k
\vec{b} = 2i + 2j + k
b
=
2
i
+
2
j
+
k
and
d
⃗
=
a
⃗
×
b
⃗
\vec{d} = \vec{a} \times \vec{b}
d
=
a
×
b
. If
c
⃗
\vec{c}
c
is a vector such that
a
⃗
⋅
c
⃗
=
∣
c
⃗
∣
\vec{a} \cdot \vec{c} = |\vec{c}|
a
⋅
c
=
∣
c
∣
,
∥
c
⃗
−
2
d
⃗
∥
=
8
\|\vec{c} - 2\vec{d}\| = 8
∥
c
−
2
d
∥
=
8
and the angle between
d
⃗
\vec{d}
d
and
c
⃗
\vec{c}
c
is
π
4
\frac{\pi}{4}
4
π
, then
∣
10
−
3
b
⃗
⋅
c
⃗
+
∣
d
⃗
∣
∣
2
\left|10 - 3\vec{b} \cdot \vec{c} + |\vec{d}|\right|^2
10
−
3
b
⋅
c
+
∣
d
∣
2
is equal to:
JEE Main - 2025
Mathematics
Vectors
View Solution
If the components of
a
⃗
=
α
i
^
+
β
j
^
+
γ
k
^
\vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}
a
=
α
i
^
+
β
j
^
+
γ
k
^
along and perpendicular to
b
⃗
=
3
i
^
+
j
^
−
k
^
\vec{b} = 3\hat{i} + \hat{j} - \hat{k}
b
=
3
i
^
+
j
^
−
k
^
respectively are
16
11
(
3
i
^
+
j
^
−
k
^
)
\frac{16}{11} (3\hat{i} + \hat{j} - \hat{k})
11
16
(
3
i
^
+
j
^
−
k
^
)
and
1
11
(
−
4
i
^
−
5
j
^
−
17
k
^
)
\frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k})
11
1
(
−
4
i
^
−
5
j
^
−
17
k
^
)
, then
α
2
+
β
2
+
γ
2
\alpha^2 + \beta^2 + \gamma^2
α
2
+
β
2
+
γ
2
is equal to:
JEE Main - 2025
Mathematics
Vectors
View Solution
Number of functions
f
:
{
1
,
2
,
…
,
100
}
→
{
0
,
1
}
f: \{1, 2, \dots, 100\} \to \{0, 1\}
f
:
{
1
,
2
,
…
,
100
}
→
{
0
,
1
}
, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
JEE Main - 2025
Mathematics
Vectors
View Solution
Let
a
⃗
=
i
+
j
+
k
\vec{a} = i + j + k
a
=
i
+
j
+
k
,
b
⃗
=
2
i
+
2
j
+
k
\vec{b} = 2i + 2j + k
b
=
2
i
+
2
j
+
k
and
d
⃗
=
a
⃗
×
b
⃗
\vec{d} = \vec{a} \times \vec{b}
d
=
a
×
b
. If
c
⃗
\vec{c}
c
is a vector such that
a
⃗
⋅
c
⃗
=
∣
c
⃗
∣
\vec{a} \cdot \vec{c} = |\vec{c}|
a
⋅
c
=
∣
c
∣
,
∥
c
⃗
−
2
d
⃗
∥
=
8
\|\vec{c} - 2\vec{d}\| = 8
∥
c
−
2
d
∥
=
8
and the angle between
d
⃗
\vec{d}
d
and
c
⃗
\vec{c}
c
is
π
4
\frac{\pi}{4}
4
π
, then
∣
10
−
3
b
⃗
⋅
c
⃗
+
∣
d
⃗
∣
∣
2
\left|10 - 3\vec{b} \cdot \vec{c} + |\vec{d}|\right|^2
10
−
3
b
⋅
c
+
∣
d
∣
2
is equal to:
JEE Main - 2025
Mathematics
Vectors
View Solution
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