Vector Algebra

Let $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ be three vectors. If $\mathbf...

$P$ is the circumcentre of $\triangle A B C$. If the position vectors of $A, B, C$ and $P$ are $\mathbf{a}, \mathbf{b}, \mathbf{c}, \frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{4}$ respectively, then the position vector of the orthocentre of thi...

The set of all real values of $c$ so that the angle between the vectors $\mathbf{a}=c x \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{b}=x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 c x \hat{\mathbf{k}}$ is an obtuse angle...

If $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are non-coplanar vectors and $p, q$ are real numbers, then the equality $[3 \mathbf{u} p \mathbf{v} p \mathbf{w}]-[p \mathbf{v} \mathbf{w} q \mathbf{u}]-[2 \mathbf{w} q \mathbf{v} q \mathbf{u}]=0$ hol...

If the position vectors of $A, B, C, D$ are $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ resp...

If $\mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c}$ are three units vectors such that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{\mathbf{c}}{\mathbf{2}}$ and $\alpha, \beta$ are the angles between $\m...

Let $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ be the position ...

If $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are two vectors such that $\mathbf{r} \times \mathbf{a}=\mathbf{b} \times \mathbf{a} \cdot \mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mat...

If $P=(\mathbf{a} \times \hat{\mathbf{i}})^2+(\mathbf{a} \times \hat{\mathbf{j}})^2+(\mathbf{a} \times \hat{\mathbf{k}})^2$ and $Q=(\mathbf{a} \cdot \hat{\mathbf{i}})^2+(\mathbf{a} \cdot \hat{\mathbf{j}})^2+(\mathbf{a} \cdot \hat{\mathbf{k}...

$\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{c}$ are the position vectors of three non-collinear points on a plane. If
$$ \alpha=[\mathbf{a b c}] \text { and } \mathbf{r}=\mathbf{a} \times \mathbf{b}-\mathbf{c} \times \mathbf{b}-\mathbf{a} \t...

$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are three vectors. If $\mathbf{r}$ is a ve...

Let $(x, y) \in R \times R$ and $\mathbf{a}=x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=6 \hat{\mathbf{i}}-y\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ be two vectors. If
$$ |\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot...

If the vectors $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $p \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are coplanar, then the unit vector in the directi...

Let $\mathbf{a}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{b}$ be two perpendicular vectors in the $X O Y$-plane. A vector $\mathbf{c}$ in the same plane and having projections 1 and 2 respectively on $\mathbf{a}$ and $\mathbf{b}$ ...

If the position vectors of the vertices $A, B, C$ of a triangle are $3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 5(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$ respec...

In a right angled triangle, if the position vector of the vertex having the right angle is $-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the position vector of the mid-point of its hypotenuse is $6 \hat{\mathbf{i}}+2 \hat{...

If the vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+l \hat{\mathbf{k}},-3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-4 l \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 / \hat{\mathbf{k}}$ form a right-angled triangle for a positive v...

If the magnitudes of $\mathbf{a}, \mathbf{b}$ and $\mathbf{a}+\mathbf{b}$ are respectively 3,4 and 5 , then the magnitude of $\mathbf{a}-\mathbf{b}$ is

If $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are the position vectors of four poin...

A line segment $P Q$ has the length 63 and direction ratios $(3,-2,6)$. If this line makes an obtuse angle with $X$-axis, then the components of the vector $\mathbf{P Q}$ are

A unit vector that is perpendicular to the vector $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and coplanar with the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\m...

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

If $\bar{a}$ is a unit vector, then
\(|\mathbf{a} \times \hat{\mathbf{i}}|^2+|\mathbf{a} \times \hat{\mathbf{j}}|^2+|\mathbf{a} \times \hat{\mathbf{k}}|^2=\)

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=-2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{c}=5 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{d}=3 \hat{\mathbf...

$3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+5 \hat{\mathbf{j}}$ are the position vectors of three non-collinear points $A, B, C$ respectively. If the perpendicular drawn from ...

If $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{b}=-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are two vectors, then the vector of magnitude 28 units in the direction of the vector $\mathbf...

If $A=(0,4,-3), B=(5,0,12)$ and $C=(7,24,0)$, then $\sqrt{B A C}=$

If $\mathbf{a}=\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}, \mathbf{b}=-2 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ are three vectors such th...

For $a \in R$, if the vectors $\mathbf{p}=(a+1) \hat{\mathbf{i}}+a \hat{\mathbf{j}}+a \hat{\mathbf{k}}$, $\mathbf{q}=a \hat{\mathbf{i}}+(a+1) \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ and $\mathbf{r}=a \hat{\mathbf{i}}+a \hat{\mathbf{j}}+(a+1) \...

Points $P$ and $Q$ are given by $\mathbf{O P}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{O Q}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. A line along the vector $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$...

\((\mathbf{a}+2 \mathbf{b}-\mathbf{c}) \cdot(\mathbf{a}-\mathbf{b}) \times(\mathbf{a}-\mathbf{b}-\mathbf{c})=\)

Let $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ be two vectors. If $\mathbf{c}^{\text {is }}$ vector such that $\mathbf{a} \cdot \mathbf{c}=|\mathbf{c}|,|\mathbf{c}-...

If $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+6 \hat{\mathbf{k}} ; \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$, then $\mathbf{a} \times \mathbf{b} \times \mathbf{b...

If the vector $\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is along the internal bisector of the angle between the vectors $\mathbf{a}$ and $-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the unit vector along $\ma...

The points in the argand plane represented by the complex numbers $4 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ ...

Let the position vectors of the vertices of a $\triangle A B C$ be $\mathbf{a , b}, \mathbf{c}$. If on the plane of the triangle, $P$ is a point having position vector $\mathbf{x}$ such that $\mathbf{x} \cdot(\mathbf{c}-\mathbf{b})=\mathbf{...

If $O(0,0,0), A(1,2,1), B(2,1,3)$ and $C(-1,1,2)$ are the vertices of a tetrahedron, then the acute angle between its face $O A B$ and edge $B C$ is

If the points $A, B, C, D$ with positions vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\math...

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors such that $|\mathbf{a}|=2,|\mathbf{b}|=3$, $|\mathbf{c}|=5,|\mathbf{a}+\mathbf{b}+\mathbf{c}|=\sqrt{69}$. If $(\mathbf{a} \cdot \mathbf{b})=(\mathbf{b} \cdot \mathbf{c})=\frac{\pi}{3}$,...

If the angles between the sides of the $\triangle A B C$ formed by $A(2,3,5), B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$, then $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=$

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors. If $\mathbf{a}, \mathbf{b}$ are perpendicular vectors, $(\mathbf{a}-\mathbf{c}) \cdot(\mathbf{b}+\mathbf{c})=0$ and $\mathbf{c}=l \mathbf{a}+m \mathbf{b}+n(\mathbf{a} \times \mathbf{b})...

$\mathbf{a}, \mathbf{b}$ are position vectors of the point $A$ and $B$ respectively, $C$ and $D$ are points on the line $A B$ such that $\mathbf{A B}, \mathbf{A C}$ and $\mathbf{B D}, \mathbf{B A}$ are two pairs of like vectors. If $\mathbf...

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three unit vectors such that $|\mathbf{a}-\mathbf{b}|^2+|\mathbf{b}-\mathbf{c}|^2+|\mathbf{c}-\mathbf{a}|^2=15$, then $|\mathbf{a}-\mathbf{b}-\mathbf{c}|^2-4(\mathbf{b} \cdot \mathbf{c})=$

If $\mathbf{a}=\hat{\mathbf{i}}+p \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=p \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are three vectors such that $|\mat...

If $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$, $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{d}=\hat{\mathbf{i}}+...

Let $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}},-13 \hat{\mathbf{i}}-11 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ be the position vectors of three points. $A, B$ and $C$, respecti...

If $\mathbf{a}=-4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{b}=\sqrt{2} \hat{\mathbf{i}}-\sqrt{2} \hat{\mathbf{j}}$ are two vectors, then angle between the vectors $2 \mathbf{a}$ and $\frac{\mathbf{b}}{2}$ is

If the vectors $a \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+b \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+c \hat{\mathbf{k}}$ $(a \neq b \neq c \neq 1)$ are coplanar, then $\frac{1}{1...

A unit vector perpendicular to the vectors $a=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{b}=3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is

The orthogonal projection vector of $a=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ on $\mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ is