1
WB JEE 2024
+1
-0.25

A unit vector in XY-plane making an angle $$45^{\circ}$$ with $$\hat{i}+\hat{j}$$ and an angle $$60^{\circ}$$ with $$3 \hat{i}-4 \hat{j}$$ is

A
$$\frac{13}{14} \hat{i}+\frac{1}{14} \hat{j}$$
B
$$\frac{1}{14} \hat{i}+\frac{13}{14} \hat{j}$$
C
$$\frac{13}{14} \hat{\mathrm{i}}-\frac{1}{14} \hat{\mathrm{j}}$$
D
$$\frac{1}{14} \hat{i}-\frac{13}{14} \hat{j}$$
2
WB JEE 2023
+1
-0.25

The value of 'a' for which the scalar triple product formed by the vectors $$\overrightarrow \alpha = \widehat i + a\widehat j + \widehat k,\overrightarrow \beta = \widehat j + a\widehat k$$ and $$\overrightarrow \gamma = a\widehat i + \widehat k$$ is maximum, is

A
3
B
$$-$$3
C
$$- {1 \over {\sqrt 3 }}$$
D
$${1 \over {\sqrt 3 }}$$
3
WB JEE 2023
+2
-0.5

If the volume of the parallelopiped with $$\overrightarrow a \times \overrightarrow b ,\overrightarrow b \times \overrightarrow c$$ and $$\overrightarrow c \times \overrightarrow a$$ as conterminous edges is 9 cu. units, then the volume of the parallelopiped with $$(\overrightarrow a \times \overrightarrow b ) \times (\overrightarrow b \times \overrightarrow c ),(\overrightarrow b \times \overrightarrow c ) \times (\overrightarrow c \times \overrightarrow a )$$, and $$(\overrightarrow c \times \overrightarrow a ) \times (\overrightarrow a \times \overrightarrow b )$$ as conterminous edges is

A
9 cu. units
B
729 cu. units
C
81 cu. units
D
243 cu. units
4
WB JEE 2022
+1
-0.25

If $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c$$ is unit vector perpendicular to $$\overrightarrow a$$ and coplanar with $$\overrightarrow a$$ and $$\overrightarrow b$$, then unit vector $$\overrightarrow d$$ perpendicular to both $$\overrightarrow a$$ and $$\overrightarrow c$$ is

A
$$\pm {1 \over {\sqrt 6 }}\left( {2\widehat i - \widehat j + \widehat k} \right)$$
B
$$\pm {1 \over {\sqrt 2 }}\left( {\widehat j + \widehat k} \right)$$
C
$$\pm {1 \over {\sqrt 6 }}\left( {\widehat i - 2\widehat j + \widehat k} \right)$$
D
$$\pm {1 \over {\sqrt 2 }}\left( {\widehat j - \widehat k} \right)$$
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