1
JEE Main 2021 (Online) 25th February Evening Shift
Numerical
+4
-1
Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\overrightarrow a$$ and $$\overrightarrow b$$ is $$8\sqrt 3$$ square units, then $$\overrightarrow a$$ . $$\overrightarrow b$$ is equal to __________.
2
JEE Main 2021 (Online) 25th February Morning Shift
Numerical
+4
-1
Let $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k$$ be three given vectors. If $$\overrightarrow r$$ is a vector such that $$\overrightarrow r \times \overrightarrow a = \overrightarrow c \times \overrightarrow a$$ and $$\overrightarrow r .\,\overrightarrow b = 0$$, then $$\overrightarrow r .\,\overrightarrow a$$ is equal to __________.
3
JEE Main 2021 (Online) 24th February Morning Shift
Numerical
+4
-1
Out of Syllabus
Let three vectors $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ be such that $$\overrightarrow c$$ is coplanar
with $$\overrightarrow a$$ and $$\overrightarrow b$$, $$\overrightarrow a .\overrightarrow c$$ = 7 and $$\overrightarrow b$$ is perpendicular to $$\overrightarrow c$$, where
$$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = 2\widehat i + \widehat k$$ , then the
value of $$2{\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2}$$ is _____.
4
JEE Main 2020 (Online) 6th September Evening Slot
Numerical
+4
-0
If $$\overrightarrow x$$ and $$\overrightarrow y$$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$ and $${2\overrightarrow x + \lambda \overrightarrow y }$$ is perpendicular to $${\overrightarrow y }$$, then the value of $$\lambda$$ is _________ .