1
JEE Main 2022 (Online) 27th June Morning Shift
MCQ (Single Correct Answer)
+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 Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus

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
$$- 12\overrightarrow b \,.\,\left( {\overrightarrow c \times \overrightarrow a } \right)$$
3
JEE Main 2022 (Online) 25th June Morning Shift
MCQ (Single Correct Answer)
+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
4
JEE Main 2022 (Online) 24th June Evening Shift
MCQ (Single Correct Answer)
+4
-1

Let $$\widehat a$$ and $$\widehat b$$ be two unit vectors such that $$|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$$. If $$\theta$$ $$\in$$ (0, $$\pi$$) is the angle between $$\widehat a$$ and $$\widehat b$$, then among the statements :

(S1) : $$2|\widehat a \times \widehat b| = |\widehat a - \widehat b|$$

(S2) : The projection of $$\widehat a$$ on ($$\widehat a$$ + $$\widehat b$$) is $${1 \over 2}$$

A
Only (S1) is true.
B
Only (S2) is true.
C
Both (S1) and (S2) are true.
D
Both (S1) and (S2) are false.
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