Vector Algebra

\(\vec{a}=4 \hat{i}+13 \hat{j}-18 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}+3 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}-4 \hat{k}\) are three vectors such that \(\vec{a}=x \vec{b}+y \vec{c}\), then \(x+y=\)

If the vectors \(\vec{a}=2 \hat{i}+p \hat{j}+4 \hat{k}\) and \(\vec{b}=6 \hat{i}-9 \hat{j}+q \hat{k}\) are collinear, then \(p\) and \(q\) are

If \(\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}\) are coterminus edges of a parallelopiped, then its volume is

For any non-zero vectors \(\bar{a}, \bar{b}, \bar{c}\), the value of \(\bar{a} \cdot[(\bar{b} \times \bar{c}) \times(\bar{a}+\bar{b}+\bar{c})]\) is

If \(\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}\) and \(\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}\), then which of the following is valid.

If the angle between the vectors \(\overline{\mathrm{a}}=2 \lambda^2 \hat{\mathrm{i}}+4 \lambda \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overline{\mathrm{b}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}\) is obtuse, then...

If \(3 \hat{j}, 4 \hat{k}\) and \(3 \hat{j}+4 \hat{k}\) are the position vectors of the vertices \(A, B, C\) respectively of \(\triangle A B C\), then the position vector of the point in which the bisector of \(\angle \mathrm{A}\) meets $$\...

If \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{c}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}\) and $$\overlin...

Let \(\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}\) and \(\bar{w}=\hat{i}+3 \hat{k}\). If \(\bar{u}\) is a unit vector, then the maximum value of the scalar triple product \([\bar{u} \bar{v} \bar{w}]\) is

If \(\hat{a}\) is a unit vector such that \((\bar{x}-\hat{a}) \cdot(\bar{x}+\hat{a})=8\), then \(|\bar{x}|=\)

Let \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) and \(\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}\). If \(\overline{\mathrm{c}}\) is a vector such that $$\overline{\mathrm{a}} \cdot \overline...

The projection of \(\bar{a}=\hat{i}-2 \hat{j}+\hat{k}\) on \(\bar{b}=2 \hat{i}-\hat{j}+\hat{k}\) is

If \(\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\bar{c}=3 \hat{i}+\lambda \hat{j}+5 \hat{k}\) are coplanar, then \(\lambda\) is the root of the equation

If \(\bar{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=-\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\bar{c}=\hat{i}+\hat{j}-2 \hat{k}\), then \((\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=\)

The vectors \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) are the sides of a triangle \(\mathrm{ABC}\). The length of the...

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are mutually perpendicular vectors having magnitudes \(1,2,3\) respectively, then $$\left[\begin{array}{lll}\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\ma...

If \(|\bar{u}|=2\) and \(\bar{u}\) makes angles of \(60^{\circ}\) and \(120^{\circ}\) with axes \(\mathrm{OX}\) and \(\mathrm{OY}\) in the origin, then \(\bar{u}=\)

If \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}\) with \(|\overline{\mathrm{a}}|=3,|\overline{\mathrm{b}}|=5\) and \(|\overline{\mathrm{c}}|=7\), then angle between \(\overline{\mathrm{a}}\) and $$\overli...

The area of the parallelogram whose diagonals are represented by the vectors \(\bar{a}=3 \hat{i}-\hat{j}-2 \hat{k}\) and \(\bar{b}=-\hat{i}+3 \hat{j}-3 \hat{k}\) is

The position vector of the point of inersection of the medians of a triangle, whose vertices are \(A(1,2,3), B(1,0,3)\) and \(C(4,1,-3)\) is

The vector equation of the line whose Cartesian equations are \(y=2\) and \(4 x-3 z+5=0\) is

If the vectors \(2 \hat{i}-\hat{j}-\hat{k} ; \hat{i}+2 \hat{j}-3 \hat{k}\) and \(3 \hat{i}+\lambda \hat{j}+5 \hat{k}\) are coplanar, then the value of \(\lambda\) is

If \(|\bar{a}|=3,|\bar{b}|=4,|\bar{a}-\bar{b}|=5\), then \(|\bar{a}+\bar{b}|=\)

If \(\bar{a}=3 \hat{i}-5 \hat{j}, \bar{b}=6 \hat{i}+3 \hat{j}\) are two vectors and \(\bar{c}\) is a vector such that \(\bar{c}=\bar{a} \times \bar{b}\), then \(a: b\) : is

If vectors \(\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}\) and \(\bar{c}=3 \hat{i}+\hat{j}+2 \hat{k}\) are such that, \(\bar{a}+\lambda \bar{b}\) is perpendicular to \(\bar{c}\), then \(\lambda=\)

In a quadrilateral PQRS, M and N are mid-points of the sides PQ and RS respectively. If \(\overline {PS} + \overline {QR} = t\overline {MN}\), then t =

The vertices of triangle \(\mathrm{ABC}\) are \(\mathrm{A} \equiv(3,0,0) ; \mathrm{B} \equiv(0,0,4) ; \mathrm{C} \equiv(0,5,4)\). Find the position vector of the point in which the bisector of angle A meets B C is

The distance between parallel lines
$$\begin{aligned}
& \bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-2 \hat{k}) \text { and } \\
& \bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+\hat{j}-2 \hat{k}) \text { is }
\end{align...

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

If \({\pi \over 2} < \theta < \pi\) and \(|\overline a | = 5,|\overline b | = 13,|\overline a \times \overline b | = 25\), then the value of \(\overline a \,.\,\overline b\) is

If \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{c}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) an...

\((2 \hat{\mathrm{i}}+6 \hat{\mathrm{i}}+27 \hat{\mathrm{k}}) \times(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\mu \hat{\mathrm{k}})=\overline{0}\), then \(\lambda\) and \(\mu\) are respectively

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}} , \overline{\mathrm{c}}\) are three vectors which are perpendicular to \(\overline{\mathrm{b}}+\overline{\mathrm{c}}, \overline{\mathrm{c}}+\overline{\mathrm{a}}\) and $$\overline{\mathrm{a}...

If \(\overline{\mathrm{e}}_1, \overline{\mathrm{e}}_2\) and \(\overline{\mathrm{e}}_1+\overline{\mathrm{e}}_2\) are unit vectors, then the angle between \(\overline{\mathrm{e}}_1\) and \(\overline{\mathrm{e}}_2\) is

If the volume of a tetrahedron whose conterminous edges are \(\vec{\mathrm{a}}+\vec{\mathrm{b}}, \vec{\mathrm{b}}+\vec{\mathrm{c}}, \vec{\mathrm{c}}+\vec{\mathrm{a}}\) is 24 cubic units, then the volume of parallelopiped whose coterminous e...

If \(\bar{r}=-4 \hat{i}-6 \hat{j}-2 \hat{k}\) is a linear combination of the vectors \(\bar{a}=-\hat{i}+4 \hat{j}+3 \hat{k}\) and \(\bar{b}=-8 \hat{i}-\hat{j}+3 \hat{k}\), then

If area of the parallelogram with \(\bar{a}\) and \(\bar{b}\) as two adjacent sides is 20 square units, then the area of the parallelogram having \(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}\) and $$2 \overline{\mathrm{a}}+3 \overline{\m...

\(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are three vectors such that \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}\) and $$|\overline{\mathrm{a}}|=3,|\overline{\mathrm...

If \([\bar{a} \bar{b} \bar{c}]=4\), then the volume (in cubic units) of the parallelopiped with \(\bar{a}+2 \bar{b}, \bar{b}+2 \bar{c}\) and \(\overline{\mathrm{c}}+2 \overline{\mathrm{a}}\) as coterminal edges, is

\(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are vectors such that \(|\overline{\mathrm{a}}|=5,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=3\) and each is perpendicular to the sum of the other two, then $$|\...

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors and $p=\frac{\mathbf{b} \times \mathbf{c}}{[a b c]}, q=\frac{\mathbf{c} \times \mathbf{a}}{[a b c]}, r=\frac{\mathbf{a} \times \mathbf{b}}{[a b c]}$, then $\mathbf{a} \cdot \m...

$\mathbf{a}$ and $\mathbf{b}$ are non-collinear vectors. If $p=(2 x+1) a-b$ and $q=(x-2) a+b$ are collinear vectors, then $x=$

If $\mathbf{a}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \hat{\mathbf{k}}$ and $\mathbf{c}=7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+23 \hat{\mathbf{k}}$ are three vectors, then which o...

In a quadrilateral \(ABCD, M\) and \(N\) are the mid-points of the sides \(A B\) and \(C D\) respectively. If \(\mathbf{A D}+\mathbf{B C}=t \mathbf{M N}\), then \(t=\)

If \([\vec{a}\ \vec{b}\ \vec{c}\ ] \neq 0\), then \(\frac{[\vec{a}\ +\vec{b}\ \vec{b}\ +\vec{c}\ \vec{c}\ +\vec{a}\ ]}{[\vec{b}\ \vec{c}\ \vec{a}\ ]}=\)

The angles between the lines
$$\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \text { and } \mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda^{\prime}...

If the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+m \hat{\mathbf{k}}\) are coplanar, then \(m=\)

For any non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\),

If the volume of the parallelopiped whose conterminus edges are along the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) is 12, then the volume of the tetrahedron whose conterminus edges are \(\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}\) ...

Let \(G\) be the centroid of a \(\triangle A B C\) and \(\mathrm{O}_{b_\theta}\) other point in that plane, then \(\mathrm{OA}+\mathrm{OB}+\mathrm{OC}+\mathrm{CG}=\)