1
JEE Main 2022 (Online) 27th June Morning Shift
+4
-1

Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the number of vectors $$\overrightarrow b$$ such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow a$$ and $$|\overrightarrow b | \in$$ {1, 2, ........, 10} is :

A
0
B
1
C
2
D
3
2
JEE Main 2022 (Online) 26th June Evening Shift
+4
-1

Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$, $$\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j + \widehat k$$ be three given vectors. Let $$\overrightarrow v$$ be a vector in the plane of $$\overrightarrow a$$ and $$\overrightarrow b$$ whose projection on $$\overrightarrow c$$ is $${2 \over {\sqrt 3 }}$$. If $$\overrightarrow v \,.\,\widehat j = 7$$, then $$\overrightarrow v \,.\,\left( {\widehat i + \widehat k} \right)$$ is equal to :

A
6
B
7
C
8
D
9
3
JEE Main 2022 (Online) 26th June Morning Shift
+4
-1

If $$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$ and $$\overrightarrow c \,.\,\overrightarrow a = 3$$, then the value of $$\left[ {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right),\,\overrightarrow b \times \left( {\overrightarrow c \times \overrightarrow a } \right),\,\overrightarrow c \times \left( {\overrightarrow b \times \overrightarrow a } \right)} \right]$$ is :

A
0
B
$$- 6\overrightarrow a \,.\,\left( {\overrightarrow b \times \overrightarrow c } \right)$$
C
$$- 12\overrightarrow c \,.\,\left( {\overrightarrow a \times \overrightarrow b } \right)$$
D
$$- 12\overrightarrow b \,.\,\left( {\overrightarrow c \times \overrightarrow a } \right)$$
4
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1

Let $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$$ $${a_i} > 0$$, $$i = 1,2,3$$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $$\overrightarrow a$$ on the vector $$3\widehat i + 4\widehat j$$ be 7. Let $$\overrightarrow b$$ be a vector obtained by rotating $$\overrightarrow a$$ with 90$$^\circ$$. If $$\overrightarrow a$$, $$\overrightarrow b$$ and x-axis are coplanar, then projection of a vector $$\overrightarrow b$$ on $$3\widehat i + 4\widehat j$$ is equal to:

A
$$\sqrt 7$$
B
$$\sqrt 2$$
C
2
D
7
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