1
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

The vector $$\overrightarrow a = - \widehat i + 2\widehat j + \widehat k$$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $$\overrightarrow b$$. Then the projection of $$3\overrightarrow a + \sqrt 2 \overrightarrow b$$ on $$\overrightarrow c = 5\widehat i + 4\widehat j + 3\widehat k$$ is :

A
$$\sqrt6$$
B
2$$\sqrt3$$
C
1
D
3$$\sqrt2$$
2
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1
Out of Syllabus

Let $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ be three non zero vectors such that $$\overrightarrow b$$ . $$\overrightarrow c$$ = 0 and $$\overrightarrow a \times (\overrightarrow b \times \overrightarrow c ) = {{\overrightarrow b - \overrightarrow c } \over 2}$$. If $$\overrightarrow d$$ be a vector such that $$\overrightarrow b \,.\,\overrightarrow d = \overrightarrow a \,.\,\overrightarrow b$$, then $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d )$$ is equal to

A
$$\frac{1}{2}$$
B
$$-\frac{1}{4}$$
C
$$\frac{1}{4}$$
D
$$\frac{3}{4}$$
3
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1

Let $$\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k$$ and $$\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k$$. Let $${\overrightarrow \beta _1}$$ be parallel to $$\overrightarrow \alpha$$ and $${\overrightarrow \beta _2}$$ be perpendicular to $$\overrightarrow \alpha$$. If $$\overrightarrow \beta = {\overrightarrow \beta _1} + {\overrightarrow \beta _2}$$, then the value of $$5{\overrightarrow \beta _2}\,.\left( {\widehat i + \widehat j + \widehat k} \right)$$ is :

A
9
B
7
C
6
D
11
4
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that

$${{QA} \over {AR}} = {{RB} \over {BP}} = {{PC} \over {CQ}} = {1 \over 2}$$. Then $${{Area(\Delta PQR)} \over {Area(\Delta ABC)}}$$ is equal to :

A
$$\frac{5}{2}$$
B
4
C
2
D
3
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