1
JEE Main 2022 (Online) 29th July Evening Shift
Numerical
+4
-1

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$$ and $$|\vec{a} \times \vec{b}|^{2}=75$$. Then $$|\vec{a}|^{2}$$ is equal to __________.

2
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1

Let $$\overrightarrow a$$, $$\overrightarrow b$$, $$\overrightarrow c$$ be three non-coplanar vectors such that $$\overrightarrow a$$ $$\times$$ $$\overrightarrow b$$ = 4$$\overrightarrow c$$, $$\overrightarrow b$$ $$\times$$ $$\overrightarrow c$$ = 9$$\overrightarrow a$$ and $$\overrightarrow c$$ $$\times$$ $$\overrightarrow a$$ = $$\alpha$$$$\overrightarrow b$$, $$\alpha$$ > 0. If $$\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| + \left| {\overrightarrow c } \right| = {1 \over {36}}$$, then $$\alpha$$ is equal to __________.

3
JEE Main 2022 (Online) 29th June Evening Shift
Numerical
+4
-1

Let  $$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$,   $$\overrightarrow b = \widehat i + \widehat j + \widehat k$$   and   $$\overrightarrow c$$   be a vector such that   $$\overrightarrow a + \left( {\overrightarrow b \times \overrightarrow c } \right) = \overrightarrow 0$$   and   $$\overrightarrow b \,.\,\overrightarrow c = 5$$. Then the value of   $$3\left( {\overrightarrow c \,.\,\overrightarrow a } \right)$$   is equal to _________.

4
JEE Main 2022 (Online) 28th June Morning Shift
Numerical
+4
-1

If $$\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow c = {c_1}\widehat i + {c_2}\widehat j + {c_3}\widehat k$$ are coplanar vectors and $$\overrightarrow a \,.\,\overrightarrow c = 5$$, $$\overrightarrow b \bot \overrightarrow c$$, then $$122({c_1} + {c_2} + {c_3})$$ is equal to ___________.