Question

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three vectors such that $\left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=5,\overrightarrow{b}\bullet \overrightarrow{c}=10$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}.$ If $\overrightarrow{a}$ is perpendicular to the vector $\overrightarrow{b}\times \overrightarrow{c},$ then $\left|\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)\right|$ is equal to ____________.

JEE Main 2020 (09 Jan Shift 2)

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Question

If the vectors, $\overrightarrow{p}=\left(a+1\right)\widehat{i}+a\widehat{j}+a\widehat{k},\overrightarrow{q}=a\widehat{i}+\left(a+1\right)\widehat{j}+a\widehat{k}$ and $\overrightarrow{r}=a\widehat{i}+a\widehat{j}+\left(a+1\right)\widehat{k}\left(a\in R\right)$ are coplanar and $3{\left(\overrightarrow{p}.\overrightarrow{q}\right)}^2-\lambda {\left|\overrightarrow{r}\times \overrightarrow{q}\right|}^2=0$ , then the value of $\lambda$ is ________

JEE Main 2020 (09 Jan Shift 1)

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Question

Let $\overrightarrow{a}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$ and $\overrightarrow{b}=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+5\hat{\mathrm{k}}$. If $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{r},\overrightarrow{r}·\left(\alpha \hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)=3$ and $\overrightarrow{r}·(2\hat{\mathrm{i}}+5\hat{\mathrm{j}}-\alpha \hat{\mathrm{k}})=-1,\alpha \in R,$ then the value of  $\alpha +|\overrightarrow{r}{|}^2$ is equal to :

JEE Main 2021 (16 Mar Shift 2)

Options

• A: $9$
• B: $15$
• C: $13$
• D: $11$

Question

Let $\overrightarrow{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\overrightarrow{b}=7\hat{i}+\hat{j}-6\hat{k}$ If $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{r}\times \overrightarrow{b},\overrightarrow{r}·(\hat{i}+2\hat{j}+\hat{k})=-3,$ then $\overrightarrow{r}·(2\hat{i}-3\hat{j}+\hat{k})$ is equal to:

JEE Main 2021 (17 Mar Shift 1)

Options

• A: $12$
• B: $8$
• C: $13$
• D: $10$

Question

Let $\overrightarrow{\mathit{a}}=\hat{i}+2\hat{\mathit{j}}-\hat{\mathit{k}}, \overrightarrow{\mathit{b}}=\hat{\mathit{i}}-\hat{\mathit{j}}$ and $\overrightarrow{c}=\hat{\mathit{i}}-\hat{\mathit{j}}-\hat{\mathit{k}}$ be three given vectors. If $\overrightarrow{\mathit{r}}$ is a vector such that $\overrightarrow{\mathit{r}}\times \overrightarrow{\mathit{a}}=\overrightarrow{\mathit{c}}\times \overrightarrow{\mathit{a}}$ and $\overrightarrow{\mathit{r}}·\overrightarrow{b}=0,$ then $\overrightarrow{r}·\overrightarrow{a}$ is equal to

JEE Main 2021 (25 Feb Shift 1)

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Question

Let $\overrightarrow{p}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{q}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be two vectors. If a vector $\overrightarrow{r}=\left(\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}\right)$ is perpendicular to each of the vectors $(\overrightarrow{p}+\overrightarrow{q})$ and $(\overrightarrow{p}-\overrightarrow{q})$, and $\left|\overrightarrow{r}\right|=\sqrt{3}$, then $\left|\alpha \right|+\left|\beta \right|+\left|\gamma \right|$ is equal to             

JEE Main 2021 (25 Jul Shift 1)

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Question

Let $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}, a_i>0, i=1,2,3$ be a vector which makes equal angles with the coordinate axes $OX,OY$ and $OZ$. Also, let the projection of $\overrightarrow{a}$ on the vector $3\hat{i}+4\hat{j}$ be $7$ . Let $\overrightarrow{b}$ be a vector obtained by rotating $\overrightarrow{a}$ with $90°$. If $\overrightarrow{a},\overrightarrow{b}$ and $x$-axis are coplanar, then projection of a vector $\overrightarrow{b}$ on $3\hat{i}+4\hat{j}$ is equal to

JEE Main 2022 (25 Jun Shift 1)

Options

• A: $\sqrt{7}$
• B: $\sqrt{2}$
• C: $2$
• D: $7$

Question

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

JEE Main 2022 (27 Jul Shift 1)

Options

• A: $2$
• B: $\frac{39}{5}$
• C: $9$
• D: $\frac{46}{5}$

Question

Let $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{c}=2\hat{i}-3\hat{j}+2\hat{k}$. Then the number of vectors $\overrightarrow{b}$ such that $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$ and $\left|\overrightarrow{b}\right|\in \left\{1,2,\dots ,10\right\}$ is

JEE Main 2022 (27 Jun Shift 1)

Options

• A: $0$
• B: $1$
• C: $2$
• D: $3$

Question

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be the vectors along the diagonal of a parallelogram having area $2\sqrt{2}$. Let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ be acute. $\left|\overrightarrow{a}\right|=1$ and $\left|\overrightarrow{a}.\overrightarrow{b}\right|=\left|\overrightarrow{a}\times \overrightarrow{b}\right|$. If $\overrightarrow{c}=2\sqrt{2}\left(\overrightarrow{a}\times \overrightarrow{b}\right)-2\overrightarrow{b}$, then an angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

JEE Main 2022 (27 Jun Shift 2)

Options

• A: $\frac{-\pi }{4}$
• B: $\frac{5\pi }{6}$
• C: $\frac{\pi }{3}$
• D: $\frac{3\pi }{4}$

Question

Let the vectors $\overrightarrow{a}=\left(1+t\right)\hat{i}+\left(1-t\right)\hat{j}+\hat{k}$, $\overrightarrow{b}=\left(1-t\right)\hat{i}+\left(1+\mathrm{t}\right)\hat{j}+2\hat{k}$ and $\overrightarrow{c}=t\hat{i}-t\hat{j}+\hat{k}, t\in R$ be such that for $\alpha ,\beta ,\gamma \in R, \alpha \overrightarrow{a}+\beta \overrightarrow{b}+\gamma \overrightarrow{c}=\overrightarrow{0}$ $\Rightarrow \alpha =\beta =\gamma =0$. Then, the set of all values of $t$ is

JEE Main 2022 (28 Jul Shift 1)

Options

• A: a non-empty finite set
• B: equal to $N$
• C: equal to $R-\left\{0\right\}$
• D: equal to $R$

Question

Let $A, B, C$ be three points whose position vectors respectively are:

$\overrightarrow{a}=\hat{i}+4\hat{j}+3\hat{k}$

$\overrightarrow{b}=2\hat{i}+\alpha \hat{j}+4\hat{k}, \alpha \in R$

$\overrightarrow{c}=3\hat{i}-2\hat{j}+5\hat{k}$

If $\alpha$ is the smallest positive integer for which $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-collinear, then the length of the median, $△ABC$, through $A$ is:

JEE Main 2022 (29 Jun Shift 2)

Options

• A: $\frac{\sqrt{82}}{2}$
• B: $\frac{\sqrt{62}}{2}$
• C: $\frac{\sqrt{69}}{2}$
• D: $\frac{\sqrt{66}}{2}$

Question

Let $\overrightarrow{b}=\hat{i}+\hat{j}+\lambda \hat{k},\lambda \in \mathrm{ℝ}$. If $\overrightarrow{a}$ is a vector such that $\overrightarrow{a}\times \overrightarrow{b}=13\hat{i}-\hat{j}-4\hat{k}$ and $\overrightarrow{a}·\overrightarrow{b}+21=0$, then $\left(\overrightarrow{b}-\overrightarrow{a}\right)·\left(\hat{k}-\hat{j}\right)+\left(\overrightarrow{b}+\overrightarrow{a}\right)·\left(\hat{i}-\hat{k}\right)$ is equal to

JEE Main 2022 (25 Jun Shift 2)

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Question

Let $S$ be the set of all $a\in R$ for which the angle between the vectors $\overrightarrow{u}=a\left({\log }_{e}b\right)\hat{i}-6\hat{j}+3\hat{k}$ and $\overrightarrow{v}=\left({\log }_{e}b\right)\hat{i}+2\hat{j}+2a\left({\log }_{e}b\right)\hat{k},\left(b>1\right)$ is acute. Then $S$ is equal to

JEE Main 2022 (28 Jul Shift 2)

Options

• A: $\left(-\infty ,-\frac{4}{3}\right)$
• B: $\Phi$
• C: $\left(-\frac{4}{3},0\right)$
• D: $\left(\frac{12}{7},\infty \right)$

Question

Let $\overrightarrow{u}=\hat{i}-\hat{j}-2\hat{k}, \overrightarrow{v}=2\hat{i}+\hat{j}-\hat{k}$, $\overrightarrow{v}·\overrightarrow{w}=2$ and $\overrightarrow{v}\times \overrightarrow{w}=\overrightarrow{u}+\lambda \overrightarrow{v}$, then $\overrightarrow{u}·\overrightarrow{w}$ is equal to

JEE Main 2023 (24 Jan Shift 1)

Options

• A: $1$
• B: $\frac{3}{2}$
• C: $2$
• D: $-\frac{2}{3}$

Question

The vector $\overrightarrow{a}=-\hat{i}+2\hat{j}+\hat{k}$ 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 $\overrightarrow{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is

JEE Main 2023 (25 Jan Shift 1)

Options

• A: $3\sqrt{2}$
• B: $1$
• C: $\sqrt{6}$
• D: $2\sqrt{3}$

Question

Let $\mathrm{ABC}$ be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{ck}$ and $\overrightarrow{\mathrm{AC}}=6 \hat{i}+\mathrm{d} \hat{j}-2 \hat{k}, \mathrm{~d}>0$. Then the square of the length of the largest side of the triangle $\mathrm{ABC}$ is _______

JEE Main 2024 (04 Apr Shift 1)

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Question

Let $\overrightarrow{\mathrm{a}}=2 \hat{i}+5 \hat{j}-\hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-2 \hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be three vectors such that $(\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i})$. If $\vec{a} \cdot \vec{c}=-29$, then $\vec{c} \cdot(-2 \hat{i}+\hat{j}+\hat{k})$ is equal to:

JEE Main 2024 (05 Apr Shift 2)

Options

• A: 15
• B: 12
• C: 10
• D: 5

Question

Let $\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$ and $\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}$ and $\vec{r} \cdot(\vec{b}-\vec{c})=0$, then $\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}$ is equal to___________

JEE Main 2024 (08 Apr Shift 1)

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Question

Let three vectors $\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}$ form a triangle such that $\vec{c}=\vec{a}-\vec{b}$ and the area of the triangle is $5 \sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}|^2$ is equal to:

JEE Main 2024 (09 Apr Shift 1)

Options

• A: 16
• B: 14
• C: 12
• D: 10

Question

If $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k}, \overrightarrow{b}=3\left(\hat{i}-\hat{j}+\hat{k}\right)$ and $\overrightarrow{\mathrm{c}}$ be the vector such that $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}$ and $\overrightarrow{a}·\overrightarrow{c}=3$, then $\overrightarrow{a}·(\left(\overrightarrow{c}\times \overrightarrow{b}\right)-\overrightarrow{b}-\overrightarrow{c})$ is equal to

JEE Main 2024 (27 Jan Shift 1)

Options

• A: $32$
• B: $24$
• C: $20$
• D: $36$

Question

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

JEE Main 2024 (30 Jan Shift 2)

Options

• A: $90$
• B: $75$
• C: $95$
• D: $85$

Question

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

JEE Main 2024 (30 Jan Shift 2)

Options

• A: $3$
• B: $5$
• C: $1$
• D: $4$

Question

Let $\overrightarrow{a}=3\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{c}$ be a vector such that $\left(\overrightarrow{a}+\overrightarrow{b}\right)\times \overrightarrow{c}=2\left(\overrightarrow{a}\times \overrightarrow{b}\right)+24\hat{j}-6\hat{k}$ and $\left(\overrightarrow{a}-\overrightarrow{b}+\hat{i}\right).\overrightarrow{c}=-3$. Then ${\left|\overrightarrow{c}\right|}^2$ is equal to _______.

JEE Main 2024 (31 Jan Shift 2)

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Question

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\overrightarrow{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that $\overrightarrow{b}\times \overrightarrow{a}=\overrightarrow{c}\times \overrightarrow{a}$. If the angle between the vector $\overrightarrow{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$, then the greatest integer less than or equal to ${\tan }^2\theta$ is:

JEE Main 2024 (01 Feb Shift 2)

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