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Let \( \vec{x} \) be a vector in the plane containing vectors \( \vec{a}=2\hat{i}-\hat{j}+\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}-\hat{k} \). If the vector \( \vec{x} \) is perpendicular to \(( 3\hat{i}+2\hat{j}-\hat{k} )\) and its projection on \( \vec{a} \) is \( \dfrac{17\sqrt{6}}{2} \), then the value of \( |\vec{x}|^{2} \) is equal to ________.

Let the vectors
\((2 + a + b)\hat{i} + (a + 2b + c)\hat{j} - (b + c)\hat{k}\)
\((1 + b)\hat{i} + 2b\hat{j} - b\hat{k} \text{ and } (2 + b)\hat{i} + 2b\hat{j} + (1 - b)\hat{k}, \; a, b, c \in \mathbb{R}\)
Be co-planar. Then which of the following is true?

Let $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3\hat{i} + 2\hat{j} - \hat{k}$ and its projection on $\vec{a}$ is $19$ units, then $|2\vec{v}|^2$ is equal to:

A vector \( \vec{a} \) has components \( 3p \) and \( 1 \) with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, \( \vec{a} \) has components \( p + 1 \) and \( \sqrt{10} \), then a value of \( p \) is equal to:

Let \( \vec{a},\vec{b} \) and \( \vec{c} \) be three vectors such that \( \vec{a}=\vec{b}\times(\vec{b}\times\vec{c}) \). If magnitudes of the vectors \( \vec{a},\vec{b} \) and \( \vec{c} \) are \( \sqrt{2},1 \) and \( 2 \) respectively and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \theta \) \( (0<\theta<\dfrac{\pi}{2}) \), then the value of \( 1+\tan\theta \) is equal to :

If the projection of the vector \( \hat{i}+2\hat{j}+\hat{k} \) on the sum of the two vectors \( 2\hat{i}+4\hat{j}-5\hat{k} \) and \( -\lambda \hat{i}+2\hat{j}+3\hat{k} \) is \( 1 \), then \( \lambda \) is equal to ________.

$\vec{a} = \alpha\hat{i} + \beta\hat{j} + 3\hat{k}$ , $\vec{b} = -\beta\hat{i} - \alpha\hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} - \hat{k}$ such that $\vec{a} \cdot \vec{b} = 1$ and $\vec{b} \cdot \vec{c} = -3$ , then $\frac{1}{3} ((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to _____________.

Let \( O \) be the origin. Let \( \overrightarrow{OP} = x \hat{i} + y \hat{j} - \hat{k} \) and \( \overrightarrow{OQ} = -\hat{i} + 2 \hat{j} + 3x \hat{k} \), \( x, y \in \mathbb{R}, \, x > 0 \), be such that \( |\overrightarrow {PQ}| = \sqrt{20} \) and the vector \( \overrightarrow{OP} \) is perpendicular to \( \overrightarrow{OQ} \). If \( \overrightarrow{OQ} = 3 \hat{i} + z \hat{j} - 7 \hat{k} \), \( z \in \mathbb{R} \), is coplanar with \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \), then the value of \( x^{2} + y^{2} + z^{2} \) is equal to:

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

If \(|\vec{a}| = 2\), \(|\vec{b}| = 5\) and \(|\vec{a} \times \vec{b}| = 8\), then \(|\vec{a} \cdot \vec{b}|\) is equal to:

In a triangle \( ABC \), if \( | \overrightarrow{BC} | = 8, | \overrightarrow{CA} | = 7, | \overrightarrow{AB} | = 10 \), then the projection of the vector \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to:

The vector equation of the plane passing through the intersection of the planes \( \vec r \cdot (\hat i+\hat j+\hat k)=1 \) and \( \vec r \cdot (\hat i-2\hat j)=-2 \), and the point \( (1,0,2) \) is:

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

Let a vector \( \vec{a} \) be coplanar with vectors \( \vec{b}=2\hat{i}+\hat{j}+\hat{k} \) and \( \vec{c}=\hat{i}-\hat{j}+\hat{k} \). If \( \vec{a} \) is perpendicular to \( \vec{d}=3\hat{i}+2\hat{j}+6\hat{k} \), and \( |\vec{a}|=\sqrt{10} \). Then a possible value of \( [\vec{a}\ \vec{b}\ \vec{c}] + [\vec{a}\ \vec{b}\ \vec{d}] + [\vec{a}\ \vec{c}\ \vec{d}] \) is equal to:

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

For \( p>0 \), a vector \( \vec{v}_{2}=2\hat{i}+(p+1)\hat{j} \) is obtained by rotating the vector \( \vec{v}_{1}=\sqrt{3}\,p\,\hat{i}+\hat{j} \) by an angle \( \theta \) about origin in counter clockwise direction. If \( \tan\theta=\dfrac{(\alpha\sqrt{3}-2)}{(4\sqrt{3}+3)} ,\) then the value of \( \alpha \) is equal to ______.

Let \( \vec{a}, \vec{b}, \vec{c} \) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \( \theta \), with the vector \( \vec{a} + \vec{b} + \vec{c} \). Then \( 36\cos^2 2\theta \) is equal to ____________.

Let \( \vec{a} = \hat{i} + 5\hat{j} + \alpha \hat{k} \), \( \vec{b} = \hat{i} + 3\hat{j} + \beta \hat{k} \) and \( \vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k} \) be three vectors such that, \( \lvert \vec{b} \times \vec{c} \rvert = 5\sqrt{3} \) and \( \vec{a} \) is perpendicular to \( \vec{b} \). Then the greatest amongst the values of \( \lvert \vec{a} \rvert^2 \) is __________.

Let \( \vec{a}=\hat{i}+\hat{j}+\hat{k} \) and \( \vec{b}=\hat{j}-\hat{k} \). If \( \vec{c} \) is a vector such that \( \vec{a}\times \vec{c}=-\vec{b} \) and \( \vec{a}\cdot \vec{c}=3 \), then \( \vec{a}\cdot (\vec{b}\times \vec{c}) \) is equal to:

Let \( \vec{c} \) be a vector perpendicular to the vectors \( \vec{a}=\hat{i}+\hat{j}-\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}+\hat{k} \). If \( \vec{c}\cdot(\hat{i}+\hat{j}+3\hat{k})=8 \) then the value of \( \vec{c}\cdot(\vec{a}\times\vec{b}) \) is equal to ________ .

Let \( \vec{a} = \hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = -\hat{i} + 2\hat{j} + 3\hat{k} \). Then the vector product \( (\vec{a} + \vec{b}) \times \big((\vec{a} \times ((\vec{a} - \vec{b}) \times \vec{b})\times\vec b\big) \) is equal to:

In a triangle \( ABC \), if \( |\overrightarrow{BC}| = 3 \), \( |\overrightarrow{CA}| = 5 \) and \( |\overrightarrow{BA}| = 7 \), then the projection of the vector \( \overrightarrow{BA} \) on \( \overrightarrow{BC} \) is equal to:

If \( (\vec{a} + 3\vec{b}) \) is perpendicular to \( (7\vec{a} - 5\vec{b}) \) and \( (\vec{a} - 4\vec{b}) \) is perpendicular to \( (7\vec{a} - 2\vec{b}) \), then the angle between \( \vec{a} \) and \( \vec{b} \) (in degrees) is ________.

A hall has a square floor of dimension \( 10 \, \text{m} \times 10 \, \text{m} \) (see the figure) and vertical walls. If the angle \( GPH \) between the diagonals \( AG \) and \( BH \) is \( \cos^{-1}\!\left(\frac{1}{5}\right) \), then the height of the hall (in meters) is:

Let \( \vec{a} \) and \( \vec{b} \) be two non-zero vectors perpendicular to each other and \( |\vec{a}| = |\vec{b}| \). If \( |\vec{a} \times \vec{b}| = |\vec{a}| \), then the angle between the vectors \( (\vec{a} + \vec{b} +( \vec{a} \times \vec{b})) \) and \( \vec{a} \) is equal to:

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies $\vec{a} \times \{(\vec{r} - \vec{b}) \times \vec{a}\} + \vec{b} \times \{(\vec{r} - \vec{c}) \times \vec{a}\} + \vec{c} \times \{(\vec{r} - \vec{a}) \times \vec{c}\} = \vec{0}$, then $\vec{r}$ is equal to:

Let \( \vec{a}=\hat{i}+\hat{j}-\hat{k},\ \vec{b} \) and \( \vec{c}=\hat{j}-\hat{k} \) be three vectors such that \( \vec{a}\times\vec{b}=\vec{c} \) and \( \vec{a}\cdot\vec{b}=1 \). If the length of projection vector of the vector \( \vec{b} \) on the vector \( \vec{a}\times\vec{c} \) is \( l \) then the value of \( 3l^{2} \) is equal to ________.

Let \( \vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k},\ \vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k} \) and \( \vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k} \), where \( \alpha \) and \( \beta \) are integers. If \( \vec{a}\cdot\vec{b}=-1 \) and \( \vec{b}\cdot\vec{c}=10 \), then \( (\vec{a}\times\vec{b})\cdot\vec{c} \) is equal to __________.

Two vectors \( \vec{X} \) and \( \vec{Y} \) have equal magnitude. The magnitude of \( (\vec{X}-\vec{Y}) \) is \( n \) times the magnitude of \( (\vec{X}+\vec{Y}) \). The angle between \( \vec{X} \) and \( \vec{Y} \) is:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2\vec{a} + 3\vec{b}| = |3\vec{a} + \vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$. If $\frac{1}{8}\vec{a}$ is a unit vector, then $|\vec{b}|$ is equal to:

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JEE Main 2021 24th Feb Evening Shift

The vector equation of the plane passing through the intersection of the planes \( \vec r \cdot (\hat i+\hat j+\hat k)=1 \) and \( \vec r \cdot (\hat i-2\hat j)=-2 \), and the point \( (1,0,2) \) is: