Dot product of vectors practice · Easy

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

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 $|\vec{a}|=5,|\vec{b}|=4,|\vec{c}|=3$ thus what will be the value of $|\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}|$, given that $\vec{a}+\vec{b}+\vec{c}=0$

A tetrahedron has vertices at $O\;\left( {0,\;0} \right),\;A\left( {1,\;2,\;1} \right),\;B\left( {2,\;1,\;3} \right)$ and $C\left( { - 1,\;1,\;2} \right)$. Then, the angle between the faces $OAB\;\mathrm{and}\;ABC$ will be

Let the unit vectors $\mathbf{\vec a}\mathrm{\;and\;}\mathbf{\vec b}$ be perpendicular to each other and the unit vector ${\mathbf{\vec c}}$ be inclined at an angle $\mathrm{\theta }$ to both ${\mathbf{\vec a}}$ and $\mathbf{\vec b}$. If $\mathbf{\vec c} = \alpha ,\mathbf{\vec a} + \mathrm{\beta },\mathrm{\;}\mathbf{\vec b} + \gamma \left( {\mathbf{\vec a}\cdot\mathbf{\vec b}} \right),$ where $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ are scalars, then

If $\vec a,\;\,\vec b,\;\,\vec c$ are three mutually perpendicular vectors each of magnitude unity, then $\left| {\,\vec a + \vec b + \vec c\,} \right|$ is equal to

If $\mathbf{\vec a},\,\,\mathbf{\vec b}$ and $\mathbf{\vec b} + \mathbf{\vec c},\,\,\mathbf{\vec c} + \mathbf{\vec a}$ are perpendicular to $\mathbf{\vec a} + \mathbf{\vec b}$ and $\left| {\,\mathbf{\vec a} + \mathbf{\vec b}\,} \right| = 6$ respectively and if $\left| {\,\mathbf{\vec a} + \mathbf{\vec b}\,} \right| = 6$, $\left| {\,\mathbf{\vec b} + \mathbf{\vec c}\,} \right| = 8$ and $\left| {\,\mathbf{\vec c} + \mathbf{\vec a}\,} \right| = 10$, then $\left| {\,\mathbf{\vec a} + \mathbf{\vec b} + \mathbf{\vec c}\,} \right|$ is equal to

A parallelogram is constructed on the vectors $\mathbf{\vec a} = 3\vec \alpha - \mathrm{\vec \beta },\;\mathbf{\vec b} = \mathrm{\vec \alpha } + 3\mathrm{\vec \beta }$, if $\left| {\mathrm{\vec \alpha }} \right| = \left| {\mathrm{\vec \beta }} \right| = 2$ and angle between ${\mathrm{\vec \alpha }}$ and $\mathrm{\vec \beta }\;\mathrm{is}\frac{\mathrm{\pi }}{3},$ then length of diagonal of the parallelogram is

A vector $\vec{a}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i}, \hat{i}+\hat{j}$ and the plane determined by the vectors $\hat{i}-\hat{j}, \hat{i}+\hat{k}$. The obtuse angle between $\vec{a}$ and the vector $\vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}$ is

If ${{\hat n}_1},\;{{\hat n}_2}$ are two unit vectors and $\theta$ is the angle between them, then $\cos \theta /2 =$

If $\left| {\;\mathbf{\vec a}} \right| = 3,\mathrm{\;}\left| {\mathbf{\vec b}} \right| = 4$, then a value of $\lambda$ for which $\mathbf{\vec a} + \mathrm{\lambda }\mathbf{\vec b}$ is perpendicular to $\mathbf{\vec a} - \mathrm{\lambda }\mathbf{\vec b}$, is

If lines $x = ay + b,z = cy + d\mathrm{ }$ and $x = a'z + b',y = c'z + d'$ are perpendicular, then

If $|\mathbf{a}+\mathbf{b}|>|\mathbf{a}-\mathbf{b}|$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors such that $\mathbf{a}+2 \mathbf{b}$ and $5 a-4 b$ are perpendicular to each other, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

If $\mathbf{a}=(1,-1,2), \mathbf{b}=(-2,3,5), \mathbf{c}=(2,-2,4)$ and $\mathbf{i}$ is the unit vector in the $x$-direction, then $(\mathbf{a}-2 \mathbf{b}+3 \mathbf{c}) . \mathbf{i}=$

If $a \mathbf{i}+6 \mathbf{j}-\mathbf{k}$ and $7 \mathbf{i}-3 \mathbf{j}+17 \mathbf{k}$ are perpendicular vectors, then the value of $a$ is

If $O A B C$ is a tetrahedron such that $\mathrm{OA}^2+\mathrm{BC}^2$ $=\mathrm{OB}^2+\mathrm{CA}^2=\mathrm{OC}^2+\mathrm{AB}^2$, then which of the following is not true?