JEE MAIN Vector Previous Year Questions (PYQs) – Page 13 of 17

If the vectors $\vec a=\lambda\,\hat i+\mu\,\hat j+4\,\hat k$, $\vec b=-2\,\hat i+4\,\hat j-2\,\hat k$ and $\vec c=2\,\hat i+3\,\hat j+\hat k$ are coplanar and the projection of $\vec a$ on the vector $\vec b$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda+\mu$ is equal to:

Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $\overrightarrow{(AB-BC)}+\overrightarrow{(AD-DC)}=k\,\overrightarrow{FE}$, then $k$ is equal to:

Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line \[ \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}. \] Then, $\,2\alpha+3\beta+4\gamma\,$ 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 :

Let $\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$, $\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$ and $\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$ where $\alpha ,\,\beta \in R$, be three vectors. If the projection of $\overrightarrow a $ on $\overrightarrow c $ is ${{10} \over 3}$ and $\overrightarrow b \times \overrightarrow c = - 6\widehat i + 10\widehat j + 7\widehat k$, then the value of $\alpha + \beta $ is equal to :

Between the following two statements: Statement I: Let $\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a} \times \vec{r} = \vec{a} \times \vec{b}$ and $\vec{a} \cdot \vec{r} = 0$ is of magnitude $\sqrt{10}$. Statement II: In a triangle $ABC$, $\cos 2A + \cos 2B + \cos 2C \geq -\dfrac{3}{2}$.

If a unit vector $\vec{a}$ makes angles $\dfrac{\pi}{3}$ with $\hat{i}$, $\dfrac{\pi}{4}$ with $\hat{j}$ and $\theta\in(0,\pi)$ with $\hat{k}$, then a value of $\theta$ is:

If $\vec a=\hat i+2\hat k$, $\vec b=\hat i+\hat j+\hat k$, $\vec c=7\hat i-3\hat j+4\hat k$, $\ \ \vec r\times\vec b+\vec b\times\vec c=\vec 0$ and $\vec r\cdot\vec a=0$. Then $\ \vec r\cdot\vec c$ is equal to:

Let $S$ be the set of all values of $\lambda$ for which the shortest distance between the lines $\dfrac{x-\lambda}{0}=\dfrac{y-3}{4}=\dfrac{z+6}{1}$ and $\dfrac{x+\lambda}{3}=\dfrac{y}{-4}=\dfrac{z-6}{0}$ is $13$. Then $8\Big|\displaystyle\sum_{\lambda\in S}\lambda\Big|$ is equal to:

Let \(\vec{a} = 4\hat{i} + 3\hat{j}\) and \(\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}\). If \(\vec{c}\) is a vector such that \[ \vec{c}\cdot(\vec{a}\times\vec{b}) + 25 = 0,\qquad \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k}) = 4, \] and the projection of \(\vec{c}\) on \(\vec{a}\) is \(1\), then the projection of \(\vec{c}\) on \(\vec{b}\) equals: