Vector Algebra

If $\mathbf{A B}=2 \mathbf{i}+3 \mathbf{j}-6 \mathbf{k}, \mathbf{B C}=6 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k}$ are the vectors along two sides of a $\triangle A B C$. Then, perimeter of $\triangle A B C$ is

If $\theta$ is the angle between $\hat{\mathbf{f}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\hat{\mathbf{g}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ and $\sin \theta=\sqrt{\frac{24}{28}}$, then $7 a^2+...

If $\hat{\mathbf{f}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{g}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$, then the projection vector of $\hat{\mathrm{f}}$ on $\hat{\mathrm{g}}$ is

In $\triangle P Q R,(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}),(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ and $(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \mathbf{k})$are$\mathbf{}$ the position vectors of the ve...

Let $\hat{\mathbf{a}}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}, \hat{\mathbf{b}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. The projection d the sum of the vectors $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ on t...

The values of $x$ for which the angle between the vectors $x^2 \hat{\mathbf{i}}+2 x \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+x \hat{\mathbf{k}}$ is obtuse lie in the interval

Let $|\hat{\mathbf{a}}|=2=|\hat{\mathbf{b}}|=3$ and the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ be $\frac{\pi}{3}$.
If a parallelogram is constructed with adjacent sides $2 \hat{\mathbf{a}}+3 \hat{\mathbf{b}}$ and $\hat{\mat...

If the vectors $a \hat{\mathbf{i}}+\mathbf{j}+3 \hat{\mathbf{k}}, 4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ are coplanar, then $a$ is equal to

$\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are two vectors and $\mathbf{c}$ is a unit vectors lying in the plane of $\mathbf{a}$ and $\mathbf{b}$. If $\ma...

If $\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{c}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$. $\mathbf{d}=2 \hat{\mathbf{i}}+\hat{\mat...

$\mathbf{a}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \quad \mathbf{b}=\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ ar linearly dependent vectors and ...

$\mathbf{c}$ is a vector along the bisector of the internal angle between the vectors $\mathbf{a}=4 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\mathbf{b}=12 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. If the m...

If $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are position vectors of 4 points such that $2 a+3 b+5 c-10 d=0$, then the ratio in which the line joining $c$ and $d$ divides the line segment joining $a$ and $\mathbf{b}$ is

If $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}},-3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of three points, $A, B, C$ respectively, t...

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are 3 vectors such that $|\mathbf{a}|=5,|\mathbf{b}|=8,|\mathbf{c}|=11$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$, then the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors. If $\alpha \mathbf{d}=\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $\beta \mathbf{a}=\mathbf{b}+\mathbf{c}+\mathbf{d}$, then $|\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}|=$

$\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ are three unit vectors. Let $\hat{\mathbf{p}}=\hat{\mathbf{u}}+\hat{\mathbf{v}}+\hat{\mathbf{w}} \cdot \hat{\mathbf{q}}=\hat{\mathbf{u}} \times(\hat{\mathbf{v}} \times \hat{\mathbf{w}})$. If $\hat{\...

If $\mathbf{a}$ and $\mathbf{b}$ are the two non collinear vectors, then $|\mathbf{b}|\mathbf{a}+|\mathbf{a}| \mathbf{b}$ represents

If the points having the position vectors $-i+4 j-4 k_{\text {, }}$, $3 i+2 j-5 k,-3 i+8 j-5 k$ and $-3 i+2 j+\lambda k$ are coplanar, then $\lambda=$

The angle between the diagonals of the parallelogram whose adjacent sides are $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors such that $|\mathbf{a}|=|\mathbf{b}|=|\mathbf{c}|=\sqrt{3}$ and $(a+b-c)^2+(b+c-a)^2+(c+a-b)^2=36$, then $|2 a-3 b+2 c|=$

If $|f|=10,|g|=14$ and $|f-g|=15$, then $|f+g|=$

If $L M N$ are the mid-points of the sides $P Q, Q R$ and $R P d$ $\triangle P Q R$ respectively, then

$$ \begin{aligned} & \mathbf{Q M}+\mathbf{L N}+\mathbf{M L}+\mathbf{R N}-\mathbf{M N}-\mathbf{Q L}= \end{aligned} $$

Let $\mathbf{a}$ and $\mathbf{b}$ be two non-collinear vector of unit modulus. If $\mathbf{u}=\mathbf{a}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{b}$ and $\mathbf{v}=\mathbf{a} \times \mathbf{b}$, then $|\mathbf{v}|=$

Let $A B C$ be an equilateral triangle of side a. $M$ and $N$ are two points on the sides $A B$ and $A C$, respectively such that $\mathbf{A N}={ }^{\prime} K \mathbf{A C}$ and $\mathbf{A B}=3 \mathbf{A M}$. If the vectors $\mathbf{B N}$ an...

Let $\mathbf{a} \times \mathbf{b}=7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\mathbf{a}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. If the length of projection of $\mathbf{b}$ on $\mathbf{a}$ is

$$
\frac{8}...

In a regular hexagon $A B C D E F, \mathbf{A B}=\mathbf{a}$ and $\mathbf{B C}=\mathbf{b}$, then $F A=$

If $(\alpha, \beta, \gamma)$ are the direction cosines of an angular bisector of two lines whose direction ratios are $(2,2,1)$ and $(2,-1,-2)$, then $(\alpha+\beta+\gamma)^2=$

If the vectors $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$, $\mathbf{c}=3 \hat{\mathbf{i}}+p \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ are coplanar, then $p...

Let $\mathbf{a}, \mathbf{b}$ be two unit vectors. If $\mathbf{c}=\mathbf{a}+2 \mathbf{b}$ and $\mathbf{d}=5 \mathbf{a}-4 \mathbf{b}$ are perpendicular to each other, then the angle between $a$ and $b$ is

If $\mathbf{f}, \mathbf{g}, \mathbf{h}$ be mutually orthogonal vectors of equal magnitudes, then the angle between the vectors $\mathbf{f}+\mathbf{g}+\mathbf{h}$ and $\mathbf{h}$ is

If $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}, \mathbf{b}=3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{c}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ are three vectors, then a vector which is perpendicular to $\mathbf{a}$ and $\mat...

$3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}},-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of the vertices $A, B$ and $C$ of a $\tr...

Let $\mathbf{O A}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{O B}=\hat{\mathbf{i}}-4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{O C}=-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ be the position vector...

a and b are two vectors such that a is not parallel to b. If $\mathbf{p}=(x+2 y+3) \mathbf{a}+(5 x-y+2) \mathbf{b}$ and $\mathbf{q}=(2 x+3 y+5) \mathbf{a}+(x-5 y-2) \mathbf{b}$ are two vectors such that $\mathbf{p}=2 \mathbf{q}$, then $x-2 ...

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $|\mathbf{a}|=|\mathbf{b}|=\sqrt{14}$ and $\mathbf{a} \cdot \mathbf{b}=-7$, then $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \cdot \mathbf{b}|}=$

Let $\mathbf{O A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{O B}=-2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ be the position vectors of two points $A$ and $B$. If $C$ is a point on the bisector $\an...

If $7 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is the position vector of the vertex $A$ of a tetrahedron $A B C D$ and $-\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ is the position vector of the centroid of the $\...

Let $(\mathbf{a}, \mathbf{b})$ denote the angle between vectors $\mathbf{a}$ and $\mathbf{b}$. If $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}, \mathbf{a} \cdot \mathbf{b}=4$ and $(\mathbf{a}, \mathbf{b})=\cos ^{-1}\...

Let \(\mathbf{F}=2 \hat{i}+2 \hat{j}+5 \hat{k}, A=(1,2,5), B=(-1,-2,-3)\) and \(\mathbf{B A} \times \mathbf{F}=4 \hat{i}+6 \hat{j}+2 \lambda \hat{k}\), then \(\lambda=\)

Let \(a, b\) and \(c\) be unit vectors such that \(a\) is perpendicular to the plane containing \(\mathbf{b}\) and \(\mathbf{c}\) and angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(\frac{\pi}{3}\). Then, $$|\mathbf{a}+\mathbf{b}+\math...

If \(\mathbf{a}\) is collinear with \(\mathbf{b}=3 \hat{i}+6 \hat{j}+6 \hat{k}\) and \(\mathbf{a} \cdot \mathbf{b}=27\), then \(|\mathbf{a}|=\)

Three vectors of magnitudes \(a, 2 a, 3 a\) are along the directions of the diagonals of 3 adjacent faces of a cube that meet in a point. Then, the magnitude of the sum of those diagonals is

a, b, c are non-coplanar vectors. If
\(\mathbf{a}+3 \mathbf{b}+4 \mathbf{c}=x(\mathbf{a}-2 \mathbf{b}+3 \mathbf{c})+y(\mathbf{a}+5 \mathbf{b}-2 \mathbf{c}) +z(6 \mathbf{a}+14 \mathbf{b}+4 \mathbf{c}) \text {, then } x+y+z=\)

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be the position vectors of the vertices of a \(\triangle A B C\). Through the vertices, lines are drawn parallel to the sides to form the \(\Delta A^{\prime} B^{\prime} C^{\prime}\). Then, the cent...

In quadrilateral \(A B C D, \mathbf{A B}=\mathbf{a}, \mathbf{B C}=\mathbf{b}\). \(\mathbf{D A}=\mathbf{a}-\mathbf{b}, M\) is the mid-point of \(B C\) and \(X\) is a point on DM such that, \(\mathbf{D X}=\frac{4}{5}\) DM.
Then, the points $$...

Let \(\mathbf{a}=x \hat{i}+y \hat{j}+z \hat{k}\) and \(x=2 y\). If \(|\mathbf{a}|=5 \sqrt{2}\) and a makes an angle of \(135^{\circ}\) with the Z-axis, then \(\mathbf{a}=\)

The vectors \(3 \mathbf{a}-5 \mathbf{b}\) and \(2 \mathbf{a}+\mathbf{b}\) are mutually perpendicular and the vectors \(a+4 b\) and \(-\mathbf{a}+\mathbf{b}\) are also mutually perpendicular, then the acute angle between \(\mathbf{a}\) and $...

\(O A B C\) is a tetrahedron. If \(D, E\) are the mid-points of \(O A\) and \(B C\) respectively, then \(\mathbf{D E}=\)

If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=0\) and \(|\mathbf{a}|=7,|\mathbf{b}|=5,|\mathbf{c}|=3\) then the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is