1
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 ___________.

2
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

Let $$\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$$, $$\lambda$$ $$\in$$ R. If $$\overrightarrow a$$ is a vector such that $$\overrightarrow a \times \overrightarrow b = 13\widehat i - \widehat j - 4\widehat k$$ and $$\overrightarrow a \,.\,\overrightarrow b + 21 = 0$$, then $$\left( {\overrightarrow b - \overrightarrow a } \right).\,\left( {\widehat k - \widehat j} \right) + \left( {\overrightarrow b + \overrightarrow a } \right).\,\left( {\widehat i - \widehat k} \right)$$ is equal to _____________.

3
JEE Main 2022 (Online) 25th June Morning Shift
Numerical
+4
-1

Let $$\theta$$ be the angle between the vectors $$\overrightarrow a$$ and $$\overrightarrow b$$, where $$|\overrightarrow a | = 4,$$ $$|\overrightarrow b | = 3$$ and $$\theta \in \left( {{\pi \over 4},{\pi \over 3}} \right)$$. Then $${\left| {\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)} \right|^2} + 4{\left( {\overrightarrow a \,.\,\overrightarrow b } \right)^2}$$ is equal to __________.

4
JEE Main 2021 (Online) 1st September Evening Shift
Numerical
+4
-1
Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. Let a vector $$\overrightarrow v$$ be in the plane containing $$\overrightarrow a$$ and $$\overrightarrow b$$. If $$\overrightarrow v$$ is perpendicular to the vector $$3\widehat i + 2\widehat j - \widehat k$$ and its projection on $$\overrightarrow a$$ is 19 units, then $${\left| {2\overrightarrow v } \right|^2}$$ is equal to _____________.