Addition of vectors practice · Mixed difficulty

The vector equation of the line joining the points $\mathbf{i}-2 \mathbf{j}+\mathbf{k}$ and $-2 \mathbf{j}+3 \mathbf{k}$ is

For a non – zero vector $a$ the set of real numbers satisfying $\left| {(5 - x)a} \right| < \left| {2a} \right|$ consists of all $x$ such that

If the vectors $6\hat i - 2\hat j + 3\hat k,\,\,2\hat i + 3\hat j - 6\hat k$ and $3\hat i + 6\hat j - 2\hat k$ form a triangle, then it is

The equation of the line passing through the points ${a_1}\mathbf{\hat i} + {a_2}\mathbf{\hat j} + {a_3}\mathbf{\hat k}\;\mathrm{and}\;{b_1}\mathbf{\hat i} + {b_2}\mathbf{\hat j} + {b_3}\mathbf{\hat k}$ is

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

The vector c, directed along the internal bisector of the angle between the vectors $a = 7\hat i - 4\hat j - 4\hat k$ and $b = - 2\widehat i - \widehat j + 2\widehat k$ with $\left| c \right| = 5\sqrt 6 ,$

If $\mathbf{\vec a},\mathbf{\vec b},\mathbf{\vec c}$ are the position vectors of the vertices of an equilateral triangle, whose orthocenter is at the origin, then

On the xy plane where O is the origin, given points, $A(1,0),\,\,B(0,1){\text{ }}\,\,{\text{and}}\,\,{\text{ }}C(1,1).$ Let P, Q and R be moving point on the line $OA,\,\,OB,\,\,OC$ respectively such that $OP = 45t(OA),\,\,\,\,OQ = 60t(OB),\,\,\,\,OR = (1 - t)(OC)$ with $t > 0.$ If the three points P, Q and R are collinear, then the value of t is equal to

If a = (2, 5) and b = (1, 4), then the vector parallel to (a + b) is

If $D,\;E$ and $F$ are respectively the mid points of $AB,\;AC\;\mathrm{and\;}BC\;\mathrm{in\;}\Delta ABC,\;\mathrm{then\;}$

$\overrightarrow {\mathbf{BE}} + \overrightarrow {\mathbf{AF}}$ is equal to

The vectors $\widehat i + 2\widehat j + 3\widehat k,\lambda \widehat i + 4\widehat j + 7\widehat k, - 3\widehat i - 2\widehat j - 5\widehat k$ are collinear, if $\lambda$ is equal to

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 $\bar{a}+2 \bar{b}-3 \bar{c}$

The position vectors of P and Q are $5\hat i + 4\hat j + a\hat k$ and $- \hat i + 2\hat j - 2\hat k,$ respectively. If the distance between them is 7, then the value of a will be

If the position vectors of three vertices A, B and C of a $\Delta ABC$ are $7\hat j + 10\hat k,\,\, - \hat i + 6\hat j + 6\hat k\,\,\,\,{\text{and}}\,\,\, - 4\hat i + 9\hat j + 6\hat k,$ respectively. The triangle is

If vectors ${a_1} = x\widehat i - \widehat j + \widehat k$ and ${a_2} = \widehat i + y\widehat j + z\widehat k$ are collinear, then a possible unit vector parallel to the vector $x\widehat i + y\widehat j + z\widehat k$ is

If the position vectors of P and Q are $(\hat i + 3\hat j - 7\hat k)\,\,{\text{and}}\,\,(5\hat i - 2\hat j + 4\hat k)$ then $\left| {PQ} \right|$ is