JEE MAIN Vector Previous Year Questions (PYQs) – Page 17 of 17

Let $P$ be the foot of the perpendicular from the point $Q(10,-3,-1)$ on the line $\dfrac{x-3}{7}=\dfrac{y-2}{-1}=\dfrac{z+1}{-2}$. Then the area of the right-angled triangle $PQR$, where $R$ is the point $(3,-2,1)$, is:

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt {13.44} $, then the standard deviation of the second sample is :

Let a unit vector which makes an angle of $60^\circ$ with $\,2\hat i+2\hat j-\hat k\,$ and an angle of $45^\circ$ with $\,\hat i-\hat k\,$ be $\vec C$. Then $\displaystyle \vec C+\Big(-\tfrac12\,\hat i+\tfrac{1}{3\sqrt2}\,\hat j-\tfrac{\sqrt2}{3}\,\hat k\Big)$ is:

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 through $A$ of $\triangle ABC$ at the point $E$. If the length of $AD$ is $\dfrac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\dfrac{\sqrt{805}}{6\sqrt{2}}$, then the position vector of $E$ is:

Let the point, on the line passing through the points $P(1,-2,3)$ and $Q(5,-4,7)$, farther from the origin and at a distance of $9$ units from the point $P$, be $(\alpha,\beta,\gamma)$. Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

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 $\cos^{-1}!\left(\dfrac{\sqrt{5}}{2\sqrt{7}}\right)$. 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 $\left|\vec{v}_1\right|^{2}+\left|\vec{v}_2\right|^{2}$ is equal to

Let $\vec a = 3\hat i + 2\hat j + 2\hat k$ and $\vec b = \hat i + 2\hat j - 2\hat k$ be two vectors. If a vector perpendicular to both the vectors $\vec a+\vec b$ and $\vec a-\vec b$ has magnitude $12$, then one such vector is:

If $\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$ and $\left| {\overrightarrow a \times \overrightarrow b } \right|$ = 8, then $\left| {\overrightarrow a .\,\overrightarrow b } \right|$ is equal to :