Dot product of vectors practice · Mixed difficulty

If two adjacent sides of two rectangles are represented by the vectors $\mathbf{p}=5 \mathbf{a}-3 \mathbf{b}, \mathbf{q}=-\mathbf{a}-2 \mathbf{b}$ and $r=-4 a-b ; s=-a+b$ respectively, then the angle between the vectors $\mathbf{x}=\frac{1}{3}(\mathrm{p}+\mathbf{r}+s)$ and $\mathrm{y}=\frac{1}{5}(\mathbf{r}+s)$

A particle is acted on by a force of 6 units in the direction 9\widehat i + 6\widehat j + 2\hat k$9\widehat \mathbf{i} + 6\widehat \mathbf{j} + 2\mathbf{\hat k}$ and is displaced from the point 3\widehat i + 4\widehat j - 15\hat k$3\widehat \mathbf{i} + 4\widehat \mathbf{j} - 15\mathbf{\hat k}$ to the point 7\widehat i - 6\widehat j + 8\hat k$7\widehat \mathbf{i} - 6\widehat \mathbf{j} + 8\mathbf{\hat k}$. The work done is

If $(\mathbf{a} \times \mathbf{b})^2+(\mathbf{a} \cdot \mathbf{b})^2=144$ and $|\mathbf{a}|=4$, then $|\mathbf{b}|=$

The vectors $\mathbf{a}$ and $\mathbf{b}$ are not perpendicular and $\mathbf{c}$ and $\mathbf{d}$ are two vectors satisfying $\mathbf{b} \times \mathbf{c}=\mathbf{b} \times \mathbf{d}$ and $\mathbf{a} \cdot \mathbf{b}=0$. Then, the vector $\mathbf{d}$ is equal to

Let $a = 2\widehat i - 3\widehat j + 4\widehat k$ and $\;b = 7\hat i\mathrm{ } + \mathrm{ }\hat j\mathrm{ } - 6\hat k.$ If $r \times a = r \times b,r.(\widehat i + 2\widehat j + \widehat k) = - 3$ then. $r(2\hat i - \mathrm{ }3\hat j + \widehat k)$ is equal to

If $\vec a,\;\vec b,\;\vec c$ are vectors such that $\vec a.\vec b = 0$ and $\vec a + \vec b = \vec c$, then

The equation of the plane perpendicular to the line

$\frac{{x - 1}}{1} = \frac{{y - 2}}{{ - 1}} = \frac{{z + 1}}{2}\mathrm{\;and\;passing\;through\;the\;point}\left( {2,3,1} \right)\;\mathrm{is\;}$

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$, whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$, is given by

Let the cosine of angle between the vectors $\mathrm{p}$ and $\mathbf{q}$ be $\lambda$ such that $2 \mathbf{p}+\mathbf{q}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\mathbf{p}+2 \mathbf{q}=\hat{\mathbf{i}}-\hat{\mathbf{j}}$, then $\lambda$ is equal to

The work done by the force $\vec F = 2\hat i - \hat j - \hat k$ in moving an object along the vector $3\hat i + 2\hat j - 5\hat k$ is

The angle $\theta$ between two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfies the relation $\cos \theta=(\mathbf{a} \times \hat{\mathbf{i}}) \cdot(\mathbf{b} \times \hat{\mathbf{i}})+(\mathbf{a} \times \hat{\mathbf{j}}) \cdot(\mathbf{b} \times \hat{\mathbf{j}})+(\mathbf{a} \times \hat{\mathbf{k}}) \cdot(\mathbf{b} \times \hat{\mathbf{k}})$ then the least value of $|\mathbf{a}|+|\mathbf{b}|$ is equal to (where $\theta \neq 90^{\circ}$ )

If three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ satisfy $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$ and $|\mathbf{a}|=3$, $|\mathbf{b}|=5,|\mathbf{c}|=7$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

The plane through the point $\left( { - 1, - 1, - 1} \right)$ and containing the line of intersection of the planes $\mathbf{\vec r}\,\, \cdot \,\,\left( {\widehat {\mathbf{\dot i}} + 3\,\widehat {\mathbf{\dot j}} - \mathbf{\hat k}} \right) = 0$ and $\mathbf{\vec r}\,\, \cdot \,\,\left( {\widehat {\mathbf{\dot j}} + 2\,\mathbf{\hat k}} \right) = 0$ is

If $\left| {\mathrm{\;}\mathbf{\vec a} \times \mathbf{\vec b}} \right| = 4\mathrm{\;and\;}\left| {\mathrm{\;}\mathbf{\vec a}\cdot\mathbf{\vec b}} \right| = 2,\mathrm{\;then\;}{\left| {\mathbf{\vec a}} \right|^2}{\left| {\mathbf{\vec b}} \right|^2}$ is equal to

Two forces whose magnitudes are $2 \mathrm{gm}$ wt, and $3 \mathrm{gm}$ wt act on a particle in the directions of the vectors $2 \hat{i}+4 \hat{j}+4 \hat{k}$ and $4 \hat{i}+4 \hat{j}+2 \hat{k}$ resepectively. If the particle is displaced from the origin to the point $(1,2,2)$, the work done is (the unit of length being $1 \mathrm{~cm}$ ) :

If sdaa $\vec{a}, \vec{b}, \vec{c}$ are vectors such that $\vec{a}+\vec{b}+\vec{c}=0$ and $|\vec{a}|=7,|\vec{b}|=5,|\vec{c}|=3$ then angle between vector $\vec{b}$ and $\vec{c}$ is

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three vectors such that $\mathbf{a}=\mathbf{b}+\mathbf{c}$ and the angle between $\mathbf{b}$ and $\mathbf{c}$ is $\frac{\pi}{2}$, then

Forces acting on a particle have magnitude 5, 3 and 1 unit and act in the direction of the vectors 6$\mathbf{\hat i} + 2\mathbf{\hat j} + 3\mathbf{\hat k}$, $3\mathbf{\hat i} - 2\mathbf{\hat j} + 6\mathbf{\hat k}$ and $2\mathbf{\hat i} - 3\mathbf{\hat j} - 6\mathbf{\hat k}$ respectively. They remain constant while the particle is displaced from the points $A\left( {2, - 1, - 3} \right)$ to $B\left( {5, - 1,1} \right)$. The work done is

If the scalar product of the vector \widehat i + \widehat j + 2\hat k $\widehat \mathbf{i} + \widehat \mathbf{j} + 2\mathbf{\hat k}\;$ with the unit vector along m\widehat i + 2\widehat j + 3\hat k$m\widehat \mathbf{i} + 2\widehat \mathbf{j} + 3\mathbf{\hat k}$ is equal to 2, then one of the value of $m$ is

Let three vectors $a$, $b$ and $c$ be such that $a \times b = c,\,\,b \times c = a$ and $\left| a \right| = 2$. Then, which one of the following is not true?

If ${a}={i}+{j}+{k}$ and ${b}={i}-{j}+2 {k}$ then the projection of ${a}$ on ${b}$ is given by

If forces of magnitudes 6 and 7 units acting in the directions $\hat i - 2\hat j + 2\hat k$ and $2\hat i - 3\hat j - 6\hat k$ respectively act on a particle which is displaced from the point $P\left( {2,\; - 1,\; - 3} \right)$ to $Q\left( {5,\; - 1,\;1} \right)$, then the work done by the forces is

Let $\bar{u}, \bar{v}, \bar{w}$ be such that $|\bar{u}|=1,|\bar{v}|=2,|\bar{w}|=3$. If the projection $\bar{v}$ along $\bar{u}$ is equal to that of $\bar{w}$ along $\bar{u}$ and $\bar{v}, \bar{w}$ are perpendicular to each other then $|\bar{u}-\bar{v}+\bar{w}|$ equals

If $a$ is a real constant and $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are variable angles and $\sqrt{a^2-4} \operatorname{Tan} A+a \operatorname{Tan} B+\sqrt{a^2+4} \operatorname{Tan} C=6 a$ Then the least value of $\operatorname{Tan}^2 A+\operatorname{Tan}^2 B+\operatorname{Tan}^2 C=$

Suppose \vec a = λ \widehat i - 7\widehat j + 3\hat k$\mathbf{\vec a} = \lambda \;\widehat \mathbf{i} - 7\widehat \mathbf{j} + 3\mathbf{\hat k}$, \vec b = λ \widehat i + \widehat j + 2λ \hat k.$\mathbf{\vec b} = \lambda \;\widehat \mathbf{i} + \widehat \mathbf{j} + 2\lambda \;\mathbf{\hat k}.$ If the angle between ${\mathbf{\vec a}}$ and ${\mathbf{\vec b}}$ is grater than $90^\circ$, then λ satisfies the inequality

\vec a·\widehat i = \vec a·( 2\widehat i + \widehat j ) = \vec a·( \widehat i + \widehat j + 3\hat k ) = 1, $\mathbf{\vec a}\cdot\widehat \mathbf{i} = \mathbf{\vec a}\cdot\left( {2\widehat \mathbf{i} + \widehat \mathbf{j}} \right) = \mathbf{\vec a}\cdot\left( {\widehat \mathbf{i} + \widehat \mathbf{j} + 3\mathbf{\hat k}} \right) = 1,\;$ then $\mathbf{\vec a}$ is equal to

If ${\vec a}$ and ${\vec b}$ are unit vectors and $\theta$ is the angle between them then $\left| {\frac{{\vec a - \vec b}}{2}} \right|$, is

Given four vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$. The vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coplanar but not collinear pair by pair and the vector $\mathbf{d}$ is not coplanar with the vecotrs $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$. If it is known that the angle between $\mathbf{a}$ and $\mathbf{b}$ is equal to that between $b$ and c, each being equal to $60^{\circ}$. The angle between $d$ and $\mathbf{a}$ is $\alpha$ and between $d$ and $b$ is $\beta$. Then, the angle between the vectors $d$ and $c$

If $\mathbf{\vec a} = \left( {\mathbf{\hat i} + \mathbf{\hat j} + \mathbf{\hat k}} \right),\;\,\mathbf{\vec a} \cdot \mathbf{\vec b} = 1\mathrm{\;}\,\,\mathrm{and}\,\mathrm{\;}\mathbf{\vec a} \times \mathbf{\vec b} = \mathbf{\hat j} - \mathbf{\hat k},\mathrm{\;\;}\,\mathrm{then\;}\,\,\mathbf{\vec b}$ is

If $l \mathbf{a}+m \mathbf{b}+n \mathbf{c}=\mathbf{0}$, where $l, m, n$ are scalars and $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are mutually perpendicular vectors, then