1
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

The distance of the point $$Q(0,2,-2)$$ form the line passing through the point $$P(5,-4, 3)$$ and perpendicular to the lines $$\vec{r}=(-3 \hat{i}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+5 \hat{k}), \lambda \in \mathbb{R}$$ and $$\vec{r}=(\hat{i}-2 \hat{j}+\hat{k})+\mu(-\hat{i}+3 \hat{j}+2 \hat{k}), \mu \in \mathbb{R}$$ is :

A
$$\sqrt{74}$$
B
$$\sqrt{86}$$
C
$$\sqrt{54}$$
D
$$\sqrt{20}$$
2
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$$. Let a vector $$\vec{b}$$ be such that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\frac{\pi}{4}$$ and $$|\vec{b}|^2=6$$. If $$\vec{a} \cdot \vec{b}=3 \sqrt{2}$$, then the value of $$\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2$$ is equal to

A
85
B
90
C
75
D
95
3
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}|=1$$ and $$|\vec{b} \times \vec{a}|=2$$. Then $$|(\vec{b} \times \vec{a})-\vec{b}|^2$$ is equal to

A
1
B
3
C
5
D
4
4
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

Let $$\overrightarrow{\mathrm{a}}=\mathrm{a}_1 \hat{i}+\mathrm{a}_2 \hat{j}+\mathrm{a}_3 \hat{k}$$ and $$\overrightarrow{\mathrm{b}}=\mathrm{b}_1 \hat{i}+\mathrm{b}_2 \hat{j}+\mathrm{b}_3 \hat{k}$$ be two vectors such that $$|\overrightarrow{\mathrm{a}}|=1, \vec{a} \cdot \vec{b}=2$$ and $$|\vec{b}|=4$$. If $$\vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}$$, then the angle between $$\vec{b}$$ and $$\vec{c}$$ is equal to:

A
$$\cos ^{-1}\left(-\frac{1}{\sqrt{3}}\right)$$
B
$$\cos ^{-1}\left(\frac{2}{3}\right)$$
C
$$\cos ^{-1}\left(\frac{2}{\sqrt{3}}\right)$$
D
$$\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$$
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