Addition of vectors practice · Mixed difficulty

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

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=\alpha \mathbf{a}+\beta \mathbf{b}$. If the vectors, $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}$ are coplanar, then $\frac{\alpha}{\beta}$ is equal to

$ABCD$ is a quadrilateral , $P,Q$ are the mid points of $\overrightarrow {\mathbf{BC}}$ and $\overrightarrow {\mathbf{AD}}$, then $\overrightarrow {\mathbf{AB}} + \overrightarrow {\mathbf{DC}}$ is equal to

In a $\mathrm{\Delta }ABC$, if $\vec AB = \hat i - 7\hat j + \hat k$ and $\vec BC = 3\hat i + \hat j + 2\hat k$, then $\left| {\vec CA} \right| =$

If $\mathbf{a}=\mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{b}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}$ and $\mathbf{c}=3 \mathbf{i}+p \mathbf{j}+5 \mathbf{k}$ are coplanar then the value of $p$ will be [RPET $\left.\mathbf{1 9 8} \mathbf{8}_5, \mathbf{8 6}, \mathbf{8 8}, \mathbf{9 1}\right]$

p = 2a – 3b, q = a – 2b + c and r = -3a + b + 2c, where a, b, c being non – zero non – coplanar vectors, then the vector -2a + 3b – c is equal to

Let a, b and c be three non – zero vectors such that no two of these are collinear. If the vector a + 2b is collinear with c and b + 3c is collinear with $(\lambda$ being some non – zero scalar), then a + 2b + 6c is equal to

Let us define the length of a vector $a\hat i + b\hat j + c\hat k$ as $|a| + |b| + |c|.$ This definition coincides with the usual definition of length of a vector $a\hat i + b\hat j + c\hat k$ if and only if

Let $\mathbf{\vec a}\;,\mathbf{\vec b},,\overrightarrow {\mathbf{c}\;} \;\;$ be three non-zero vectors such that no two of these are collinear. If the vector $\mathbf{\vec a} + 2\;\mathbf{\vec b}$ is collinear with $\overrightarrow {\mathbf{c}\;} \;,$ then $\overrightarrow {\mathbf{a}\;} + 2\overrightarrow {\mathbf{b}\;} + 6\;\overrightarrow {\mathbf{c}\;} \;\;\mathrm{equals}\;\;\;$

In a triangle $A B C$, if $2 \overrightarrow{A C}=3 \overrightarrow{C B}$, then $2 \overrightarrow{O A}+3 \overrightarrow{O B}$ equals

If $\mathbf{a}=2 \mathbf{i}+\mathbf{j}-8 \mathbf{k}$ and $\mathbf{b}=\mathbf{i}+3 \mathbf{j}-4 \mathbf{k}$, then the magnitude of $\mathbf{a}+\mathbf{b}=$

If ${\vec a}$ is a vector of magnitude 50 collinear with the vector $\vec b = 6\hat i - 8\hat j - \frac{{15}}{2}\hat k$ and makes an acute angle with the positive direction of $z$-axis, then $\vec a =$

Let A and B be points with position vectors a and b with respect to the origin O. If the point C on OA is such that 2AC = CO, CD is parallel to OB and $\left| {CD} \right| = 3\left| {OB} \right|,$ then AD is equal to

If the point whose position vectors are 2\widehat i + \widehat j + \hat k$2\widehat \mathbf{i} + \widehat \mathbf{j} + \mathbf{\hat k}$, 6\widehat i - \widehat j + 2\hat k$6\widehat \mathbf{i} - \widehat \mathbf{j} + 2\mathbf{\hat k}$ and 14\widehat i - 5\widehat j + p\hat k$14\widehat \mathbf{i} - 5\widehat \mathbf{j} + p\mathbf{\hat k}$ are collinear, then the value of $p$ is

The position vectors of the points A, B and C are $\widehat i + 2\widehat j - \widehat k,\widehat i + \widehat j + \widehat k$ and $2\widehat i + 3\widehat j + 2\widehat k,$ respectively. If A is chosen as the origin, then the position vectors of B and C are

If $\hat i - 3\hat j + 5 \hat k$ bisects the angle between $a$ and $- \hat i + 2\hat j + 2\hat k,$ where $a$ is a unit vector, then

If forces $\overrightarrow{{P}}, \overrightarrow{{Q}}, \overrightarrow{{R}}$ acting at a point can be represented by the sides of a triangle taken in order, then

The non – zero vectors a, b and c are related by a = 8b and c = - 7b angle between a and c is

If the vectors a and b are linearly independent satisfying $(\sqrt 3 \tan \theta + 1)a + (\sqrt 3 \sec \theta - 2)b = 0,$ then the most general value of $\theta$ are

The position vector of midpoint lying on the line joining the points whose position vectors are \widehat i + \widehat j - \hat k$\widehat \mathbf{i} + \widehat \mathbf{j} - \mathbf{\hat k}$ and \widehat i - \widehat j + \hat k,$\widehat \mathbf{i} - \widehat \mathbf{j} + \mathbf{\hat k},$ is

The two vectors $\vec a = 2\hat i + \hat j + 3\hat k,\;\vec b = 4\hat i - \lambda \hat j + 6\hat k$ are parallel if $\lambda =$

If $|\vec{a}|=4$ and $-3 \leq \lambda \leq 2$, then the range of $|\lambda \vec{a}|$ is

If $A B C D$ is a parallelogram, $\overrightarrow{A B}=2 \mathbf{i}+4 \mathbf{j}-5 \mathbf{k}$ and $\overrightarrow{A D}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$, then the unit vector in the direction of $B D$ is

Which of the following is not a unit vector for all values of $\theta$

If the diagonals of a parallelogram are $3\hat i + \hat j - 2\hat k$ and $\hat i - 3\hat j + 4\hat k$, then the lengths of its sides are

Let the position vectors of the points $A, B, C$ and $D$ be $5 \hat{i}+5 \hat{j} +2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$ and $-\hat{i}+5 \hat{j}+6 \hat{k}$. Let the set $S=\{\lambda \in \mathbb{R}$ : the points $A$, $B, C$ and $D$ are coplanar}. Then $\sum\limits_{{\lambda \in S}}(\lambda+2)^{2}$ is equal to

Let $A, B, C$ be three points whose position vectors respectively are
$\vec{a}=\hat{i}+4 \hat{j}+3 \hat{k}$

$\vec{b}=2 \hat{i}+\alpha \hat{j}+4 \hat{k}, \alpha \in \mathbb{R}$

$\vec{c}=3 \hat{i}-2 \hat{j}+5 \hat{k}$
If $\alpha$ is the smallest positive integer for which $\vec{a}, \vec{b}, \vec{c}$ are non collinear, then the length of the median, in $\triangle A B C$, through $A$ is:

In the triangle $A B C, \overrightarrow{A B}=\mathbf{a}, \overrightarrow{A C}=\mathbf{c}, \overrightarrow{B C}=\mathbf{b}$, then

A line segment has length 63 and direction ratios are 3, -2 and 6. The components of line vector are

Let P be the plane passing through the intersection of the planes $\vec r \cdot (\hat i + 3\hat j - \hat k) = 5$ and $\vec r \cdot (2\hat i - \hat j + \hat k) = 3,$ and the point $(2,1, - 2).$ Let the position vectors of the points X and Y be $\hat i - 2\hat j + 4\hat k$ and $5\hat i - \hat j + 2\hat k$ respectively. Then the points