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

Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}$$, the vector $$(x \vec{a}+y \vec{b})$$ is perpendicular to the vector $$(6 y \vec{a}-18 x \vec{b})$$. Then the value of $$|\vec{a} \times \vec{b}|$$ is equal to :

A
$$9 \sqrt{3}$$
B
$$27 \sqrt{3}$$
C
9
D
81
2
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let $$\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$$ and $$\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}$$. Then the projection of $$\vec{b}-2 \vec{a}$$ on $$\vec{b}+\vec{a}$$ is equal to :

A
2
B
$$\frac{39}{5}$$
C
9
D
$$\frac{46}{5}$$
3
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1
Out of Syllabus

$$\text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2} \text {, then }|\vec{b} \times 2 \hat{j}|$$ is equal to :

A
4
B
5
C
$$\sqrt{21}$$
D
$$\sqrt{17}$$
4
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

Let $$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$$. If the projection of $$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$$ on the vector $$-\hat{i}+2 \hat{j}-2 \hat{k}$$ is 30, then $$\alpha$$ is equal to :

A
$$\frac{15}{2}$$
B
8
C
$$\frac{13}{2}$$
D
7
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