JEE MAIN Vector Previous Year Questions (PYQs) – Page 11 of 17

The foot of the perpendicular from the point $(2,0,5)$ on the line $\dfrac{x+1}{2}=\dfrac{y-1}{5}=\dfrac{z+1}{-1}$ is $(\alpha,\beta,\gamma)$. Then, which of the following is NOT correct?

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:

Let $P(3,2,3)$, $Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle PQR$. The angle $\angle QPR$ is:

Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}\times\vec{b}=\vec{c}\times\vec{b}$ and $\vec{d}\cdot\vec{a}=24$, then $|\vec{d}|^{2}$ is equal to:

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

Let three vectors $\vec a=\alpha\hat i+4\hat j+2\hat k,;\vec b=5\hat i+3\hat j+4\hat k,;\vec c=x\hat i+y\hat j+z\hat k$ 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 $\lvert\vec c\rvert$ is equal to:

Let $\overrightarrow{OA}=\vec a,\ \overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $OA$ and $OC$, then $\dfrac{\text{area of quadrilateral }OABC}{\text{area of }S}$ is equal to:

Let $\vec{\alpha}=3\hat{i}+\hat{j}$ and $\vec{\beta}=2\hat{i}-\hat{j}+3\hat{k}$. If $\vec{\beta}=\vec{\beta}{1}-\vec{\beta}{2}$, where $\vec{\beta}{1}$ is parallel to $\vec{\alpha}$ and $\vec{\beta}{2}$ is perpendicular to $\vec{\alpha}$, then $\vec{\beta}{1}\times\vec{\beta}{2}$ is equal to:

If the vector $\vec{b} = 3\vec{j} + 4\vec{k}$ is written as the sum of a vector $\vec{b_1}$ parallel to $\vec{a} = \vec{i} + \vec{j}$ and a vector $\vec{b_2}$ perpendicular to $\vec{a}$, then $\vec{b_1} \times \vec{b_2}$ is equal to:

A vector $\overrightarrow 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, $\overrightarrow a $ has components p + 1 and $\sqrt {10} $, then the value of p is equal to :