Addition of vectors practice · Medium

If the points whose position, vectors are $3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}$, $2 i+3 \mathbf{j}-4 \mathbf{k},-\mathbf{i}+\mathbf{j}+2 \mathbf{k}$ and $4 \mathbf{i}+5 \mathbf{j}+\lambda \mathbf{k}$ lie on a plane, then $\lambda=$

The position vector of the point in which the line joining the points $\mathbf{i}-2 \mathbf{j}+\mathbf{k}$ and $3 \mathbf{k}-2 \mathbf{j}$ cuts the plane through the origin and the points $4 \mathbf{j}$ and $2 \mathbf{i}+\mathbf{k}$, is

The line through $\mathbf{i}+3 \mathbf{j}+2 \mathbf{k}$ and perpendicular to the lines $\mathbf{r}=(\mathbf{i}+2 \mathbf{j}-\mathbf{k})+\lambda(2 \mathbf{i}+\mathbf{j}+\mathbf{k})$

The unit vectors orthogonal to the vector $- \hat i + 2\hat j + 2\hat k$ and making equal angles with the $X$ and $Y$ axes is (are)

Vector coplanar with vectors $\mathbf{i}+\mathbf{j}$ and $\mathbf{j}+\mathbf{k}$ and parallel to the vector $2 \mathbf{i}-2 \mathbf{j}-4 \mathbf{k}$, is

If the position vectors of the point $A, B, C$ be $\mathbf{i}, \mathbf{j}, \mathbf{k}$ respectively and $P$ be a point such that $\overrightarrow{A B}=\overrightarrow{C P}$, then the position vector of $P$ is

The position vectors of the points $A, B, C$ and $D$ are $3 {i}-2 {j}-{k}, 2 {i}+3 {j}-4 {k},-{i}+{j}+2 {k}$ and $4 {i}+5 {j}+\lambda {k}$ respectively. If the points ${A}, {B}, {C}$ and ${D}$ are on a plane, then the value of $\lambda$ is

Let two non-collinear unit vectors ${\mathbf{\hat a}}$ and ${\mathbf{\hat b}}$ from and acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow {\mathbf{OP}}$ (where $O$ is the origin) is given by $\mathbf{\hat a}\cos t + \mathbf{\hat b}\sin t$. When $P$ is farthest form origin $O$, let $M$ be the length of $\overrightarrow {\mathbf{OP}}$ and ${\mathbf{\hat u}}$ be the unit vector along $\overrightarrow {\mathbf{OP}}$ Then,

A vector of length 3 perpendicular to each of the vectors $3 i+j-4 k$ and $6 \mathbf{i}+5 \mathbf{j}-2 \mathbf{k}$ is

If the vectors $\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}, 2 \mathbf{i}-\mathbf{j}+4 \mathbf{k}$ and $3 \mathbf{i}+2 \mathbf{j}+x \mathbf{k}$ are coplanar, then the value of $x$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two non-collinear vectors and $x \mathbf{a}+y \mathbf{b}=0$

If two points are $P(7,-5,11)$ and $Q(-2,8,13)$, then the projection of PQ on a straight line with direction cosines $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$ is

The unit vector bisecting OY and OZ is

If $A B C D$ is a parallelogram and the position vectors of $A, B$ and $C$ are $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $7 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}$, then the position vector of $D$ will be

If the four points, whose position vectors are $3 \hat{i}-4 \hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-\hat{k},-2 \hat{i}-\hat{j}+3 \hat{k}$ and $5 \hat{i}-2 \alpha \hat{j}+4 \hat{k}$ are coplanar, then $\alpha$ is equal to

A unit vector is orthogonal to $5 \hat{i}+2 \hat{j}+6 \hat{k}$ and is coplanar to $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ then the vector is

Let $\vec{p}$ and $\vec{q}$ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $|\vec{p}|=p,|\vec{q}|=q$. The points ${R}$ and $S$ divide ${PQ}$ internally and externally in the ratio $2: 3$ respectively. If $\overrightarrow{O R}$ and $\overrightarrow{O S}$ are perpendicular, then

The position vectors of the points $A$ and $B$ with respect to $O$ are $2 \hat{{i}}+2 \hat{{j}}+\hat{{k}}$ and $2 \hat{{i}}+4 \hat{{j}}+4 \hat{{k}}$. The length of the internal bisector of $\angle B O A$ of $\triangle A O B$ is

$\bar{a}=-\bar{i}+\bar{j}, \bar{b}=\bar{j}-\bar{k}, \bar{c}=\bar{i}+\bar{j}+\bar{k}$ then the unit vector in the opposite direction of $\bar{a}+2 \bar{b}-3 \bar{c}$