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Let \( A, B, C \) be three points in \( xy \)-plane, whose position vector are by \( \sqrt{3}\,\hat{i}+\hat{j} \),\( \hat{i}+\sqrt{3}\,\hat{j} \) and \( a\hat{i}+(1-a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \(\displaystyle \frac{9}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is:

Let \( P \) be the image of the point \( Q(7,-2,5) \) in the line \( L:\; \dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z}{4} \) and \( R(5,p,q) \) be a point on \( L \). Then the square of the area of \( \triangle PQR \) is \( \underline{\phantom{xxxx}} \).

Let the angle $\theta, 0 < \theta < \frac{\pi}{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be $\sin^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector $\vec{c} = 3\hat{a} + 6\hat{b} + 9\left(\hat{a} \times \hat{b}\right)$, then the value of $9(\vec{c}.\hat{a}) - 3(\vec{c}.\hat{b})$ is:

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} , \\ \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a}\cdot\vec{c} = |\vec{c}| \), \( |\vec{c} - 2\vec{a}|^{2} = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b}\cdot\vec{c}| + |\vec{d}\times\vec{c}|^{2} \) is equal to ______.

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}, \ \vec{c} = \lambda \hat{j} + \mu \hat{k} \) and \( \hat{d} \) be a unit vector such that \( \vec{a} \times \hat{d} = \vec{b} \times \hat{d} \) and \( \vec{c} \cdot \hat{d} = 1 \). If \( \vec{c} \) is perpendicular to \( \vec{a} \), then \( |3\lambda \hat{d} + \mu \vec{c}|^{2} \) is equal to _________.

Let \( \vec{a} \) and \( \vec{b} \) be the vectors of the same magnitude such that \( \dfrac{|\vec{a}+\vec{b}| + |\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}| - |\vec{a}-\vec{b}|} = \sqrt{2} + 1 \). Then \( \dfrac{|\vec{a}+\vec{b}|^{2}}{|\vec{a}|^{2}} \) is:

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to:

Let the shortest distance between the lines \( \dfrac{x-3}{3} = \dfrac{y-\alpha}{-1} = \dfrac{z-3}{1} \) and \( \dfrac{x+3}{-3} = \dfrac{y+7}{2} = \dfrac{z-\beta}{4} \) be \( 3\sqrt{30} \). Then the positive value of \( 5\alpha + \beta \) is:

Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r:1 \, (r>0) \). If \( O \) is the origin and \( {\overrightarrow {OQ} \cdot \overrightarrow{OA}} - \frac{1}{5} \left|\overrightarrow{ OP} \times \overrightarrow{ OA} \right|^{2} = 10 \), then the value of \( r \) is:

Let \( \vec{a} \) and \( \vec{b} \) be two-unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \vec{a} + 2\vec{b} \) and \( 3\vec{a} - \lambda \vec{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \( [-1, 3] \) is:

Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k}, \ \vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k}, \ \vec{c} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{d} \) be a vector such that \( \vec{b} \times \vec{d} = \vec{c} \times \vec{d} \) and \( \vec{a} \cdot \vec{d} = 4 \). Then \( \left| \left( \vec{a} \times \vec{d} \right) \right|^{2} \) is equal to __________.

Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \vec{c} \) be such that \( (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \) and \( \vec{a} \cdot \vec{c} = 3 \). If \( \vec{b} \times \vec{c} = \vec{d} \), then \( \left| \vec{a} \cdot \vec{d} \right| \) is equal to:

Let \( \vec{c} \) be the projection vector of \( \vec{b}=\lambda \hat{i}+4\hat{k},\, \lambda>0 \), on the vector \( \vec{a}=\hat{i}+2\hat{j}+2\hat{k} \). If \( |\vec{a}+\vec{c}|=7 \), then the area of the parallelogram formed by the vectors \( \vec{b} \) and \( \vec{c} \) is ______.

If \( \vec{a} \) is a nonzero vector such that its projections on the vectors $2\hat{i} - \hat{j} + 2\hat{k}, \hat{i} + 2\hat{j} - 2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along \( \vec{a} \) is:

Let the three sides of a triangle \( ABC \) be given by the vectors \( 2\hat{i}-\hat{j}+ \hat{k} \), \( \hat{i}-3\hat{j}-5\hat{k} \) and \( 3\hat{i}-4\hat{j}-4\hat{k} \). Let \( G \) be the centroid of the triangle \( ABC \). Then \( 6 \left( |\overline{AG}|^2 + |\overline{BG}|^2 + |\overline{CG}|^2 \right) \) is equal to ________________.

Let \( \vec a=2\hat i-\hat j+3\hat k \), \( \vec b=3\hat i-5\hat j+\hat k \) and \( \vec c \) be a vector such that \( \vec a\times \vec c=\vec c\times \vec b \) and \( (\vec a+\vec c)\cdot(\vec b+\vec c)=168 \). Then the maximum value of \( |\vec c|^2 \) is:

Consider two vectors \( \vec{u} = 3\hat{i} - \hat{j} \) and \( \vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}, \ \lambda > 0 \). The angle between them is given by \( \cos^{-1}\!\left(\frac{\sqrt{5}}{2\sqrt{7}}\right) \). Let \( \vec{v} = \vec{v}_1 + \vec{v}_2 \), where \( \vec{v}_1 \) is parallel to \( \vec{u} \) and \( \vec{v}_2 \) is perpendicular to \( \vec{u} \). Then the value of \( \lvert \vec{v}_1 \rvert^2 + \lvert \vec{v}_2 \rvert^2 \) is equal to:

If the component of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \(\displaystyle \frac{16}{11}(3\hat{i} + \hat{j} - \hat{k}) \) and \(\displaystyle \frac{1}{11}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Let \( A \) and \( B \) be two distinct points on the line \( L : \frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2} \). Both \( A \) and \( B \) are at a distance \( 2\sqrt{17} \) from the foot of perpendicular drawn from the point \( (1,2,3) \) on the line \( L \). If \( O \) is the origin, then \( \overrightarrow{OA}\cdot\overrightarrow{OB} \) is equal to:

Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\hat{i} + 2\hat{j} + \hat{k}$, $\hat{i} + 3\hat{j} - 2\hat{k}$ and $2\hat{i} + \hat{j} - \hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt{2}}$, then the position vector of $E$ is:

Let \( \vec{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \vec{b} = \vec{a} \times (\hat{i} - 2\hat{k}) \) and \( \vec{c} = \vec{b} \times \hat{k} \). Then the projection of \( \vec{c} - 2\hat{j} \) on \( \vec{a} \) is:

Let \( \vec{a}=\hat{i}+2\hat{j}+\hat{k} \) and \( \vec{b}=2\hat{i}+\hat{j}-\hat{k} \). Let \( \hat{c} \) be a unit vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \) and be perpendicular to \( \vec{a} \). Then such a vector \( \hat{c} \) is:

Let \( \hat{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \(\displaystyle
\cos^{-1}\!\left(-\frac{1}{3}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \hat{a} \) makes an angle of \(\displaystyle
\frac{\pi}{3} \) with the vector \( \hat{i} + \alpha \hat{j} + \hat{k} \), then the value of \( \alpha \) is:

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JEE Main 2025 2nd April Morning Shift

If \( \vec{a} \) is a nonzero vector such that its projections on the vectors $2\hat{i} - \hat{j} + 2\hat{k}, \hat{i} + 2\hat{j} - 2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along \( \vec{a} \) is: