Vector Algebra

Let \(\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}\) and \(\overrightarrow{O C}=3 \vec{b}\), where \(O\) is the origin. If the area of the parallelogram with adjacent sides \(\overrightarrow{O A}\) and $$\overrig...

Let three vectors ,\(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}\) form a triangle such that $$\vec{c}=...

Let \(\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\), for whic...

Between the following two statements:
Statement I : Let \(\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\vec{b}=2 \hat{i}+\hat{j}-\hat{k}\). Then the vector \(\vec{r}\) satisfying \(\vec{a} \times \vec{r}=\vec{a} \times \vec{b}\) and $$\vec{a...

Let \(\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}\) and \(\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that $$\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \ti...

The set of all \(\alpha\), for which the vectors \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}\) and \(\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}\) are inclined at an obtuse angle for all \(t \in \mathbb{R}\), is

Let \(\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=11 \hat{i}-\hat{j}+\hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that $$(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \tim...

Let \(\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}+3 \hat{j}-5 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=3 \hat{i}-\hat{j}+\lambda \hat{k}\) be three vectors. Let $$\overrightarrow{\mathrm...

Let \(\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}\) and a vector \(\vec{c}\) be such that \(\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}\). If $$\vec{a} \cdot \vec{c}=1...

Let \(\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}\). If \(\overrightarrow{\mathrm{c}}\) is a is vector such that $$|\overrightarrow{\mathrm{c}}| \geq 6, \overrightarrow{\mathrm{a...

Let \(\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}\). Then the square of the projection of \(\vec{a}\) on \(\vec{b}\) is:

If \(\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)\) and \(\mathrm{D}(-1,-4,-2)\) are the vertices of a quadrilateral ABCD, then its area is :

Let \(\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that $$(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \...

Consider three vectors \(\vec{a}, \vec{b}, \vec{c}\). Let \(|\vec{a}|=2,|\vec{b}|=3\) and \(\vec{a}=\vec{b} \times \vec{c}\). If \(\alpha \in\left[0, \frac{\pi}{3}\right]\) is the angle between the vectors \(\vec{b}\) and \(\vec{c}\), then ...

Let \(\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}\) and \(\vec{c}\) be three vectors such that \((\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i})\). If $$\vec{a} \cdot \vec{c}=...

Let \(\mathrm{ABC}\) be a triangle of area \(15 \sqrt{2}\) and the vectors \(\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{c} \hat{k}\) and $$\overrighta...

Let a unit vector which makes an angle of \(60^{\circ}\) with \(2 \hat{i}+2 \hat{j}-\hat{k}\) and an angle of \(45^{\circ}\) with \(\hat{i}-\hat{k}\) be \(\vec{C}\). Then $$\vec{C}+\left(-\frac{1}{2} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\fr...

For \(\lambda>0\), let \(\theta\) be the angle between the vectors \(\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). If the vectors \(\vec{a}+\vec{b}\) and \(\vec{a}-\vec{b}\) are mutually perpendicu...

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}\) and \(\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}\).
If \(\vec{d}\) is the unit vector in the direction of \(\vec{b}+\vec{c}\) such that $$\vec{a} \c...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=1,|\vec{b}|=4\), and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\alpha\), then...

The distance of the point \(Q(0,2,-2)\) form the line passing through the point \(P(5,-4, 3)\) and perpendicular to the lines \(\vec{r}=(-3 \hat{i}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+5 \hat{k}), \lambda \in \mathbb{R}\) and $$\vec{r}=(\...

Let \(\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}\) and \(\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}\) be three vectors. If a vectors \(\vec{p}\) satisfies \(\vec{p} \times \vec{b}=\vec{c} \times \vec{b}\) and $$\vec{p...

Let \(\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{c}\) be a vector such that \((\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}\) and $$(\vec{a}-\vec{b}+\hat{i}) \cdot...

Let \(\overrightarrow{\mathrm{a}}=\mathrm{a}_1 \hat{i}+\mathrm{a}_2 \hat{j}+\mathrm{a}_3 \hat{k}\) and \(\overrightarrow{\mathrm{b}}=\mathrm{b}_1 \hat{i}+\mathrm{b}_2 \hat{j}+\mathrm{b}_3 \hat{k}\) be two vectors such that $$|\overrightarro...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{b}|=1\) and \(|\vec{b} \times \vec{a}|=2\). Then \(|(\vec{b} \times \vec{a})-\vec{b}|^2\) is equal to

Let \(\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}\). Let a vector \(\vec{b}\) be such that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\) and \(|\vec{b}|^2=6\). If $$\vec{a} \cdot \vec{b}=3...

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear. If \(\vec{a}+5 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+6 \vec{c}\) is collinear with \(\vec{a}\) and $$\vec{a...

Let \(\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}\), where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quad...

Let a unit vector \(\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors $$\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}, \frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\...

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is acute, is ___________.

Let $\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k}, $
$\overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$.
Let $\overrightarrow{\mathrm{c}}$ be the vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c...

The position vectors of the vertices \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) of a triangle are \(2 \hat{i}-3 \hat{j}+3 \hat{k}, 2 \hat{i}+2 \hat{j}+3 \hat{k}\) and \(-\hat{i}+\hat{j}+3 \hat{k}\) respectively. Let \(l\) denotes the len...

Let the position vectors of the vertices \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) of a triangle be \(2 \hat{i}+2 \hat{j}+\hat{k}, \hat{i}+2 \hat{j}+2 \hat{k}\) and \(2 \hat{i}+\hat{j}+2 \hat{k}\) respectively. Let \(l_1, l_2\) and $$l_...

Let $\overrightarrow{\mathrm{a}}=-5 \hat{i}+\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-4 \hat{k}$ and
$\overrightarrow{\mathrm{c}}=(((\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \hat{i}) \ti...

Let $\overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=-\hat{i}-8 \hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}=4 \hat{i}+\mathrm{c}_2 \hat{j}+\mathrm{c}_3 \hat{k}$ be three vectors such that $\overright...

Let \(\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c}=\vec{a} \times \vec{b}\). If $$\vec{a} \cdot \vec{c}=-12, \vec{c} \cdot(\hat{i}-2 \hat...

If the points with position vectors \(\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}\) are collinear, then \((19 \alpha-6 \beta)^{2}\) is equal to :

The area of the quadrilateral \(\mathrm{ABCD}\) with vertices \(\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)\) and \(\mathrm{D}(1,-6,-7)\) is equal to :

Let the vectors \(\vec{u}_{1}=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_{2}=\hat{i}+b \hat{j}+\hat{k}\) and \(\vec{u}_{3}=c \hat{i}+\hat{j}+\hat{k}\) be coplanar. If the vectors $$\vec{v}_{1}=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \vec{v}_{2}=a \hat{i...

Let \(\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}\) and \(\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}\).

If \(\vec{d}\) is a vector perpendicular to both \(\vec{b}\) and \(\vec{c}\), and \(\vec{a} \cdot \vec{d}=18\)...

Let the position vectors of the points A, B, C and D be
\(5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(-\hat{i}+5 \hat{j}+6 \hat{k}\). Let the set $$S=\{\lambda \in \mathbb{...

The sum of all values of \(\alpha\), for which the points whose position vectors are \(\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\alpha+1) \hat{i}+2 \hat{k}\) and \(9 \hat{i}+(\alpha-8) \hat{j}+6 \hat{k}\) are coplanar, is...

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by \(\vec{a}, \vec{b}+\vec{c}\) and $$\vec{a}+2 ...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}\) and \(|\vec{a} \times \vec{b}|=\sqrt{48}\). Then \((\vec{a} \cdot \vec{b})^{2}\) is equal to ___________.

Let \(\vec{a}=2 \hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}\) and \(\vec{c}\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statements:
(A) ...

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that
$|\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$.
If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2 \pi}{3}$, then $\le...

Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}$ be three vectors. If $\vec{r}$ is a vector such

that, $\vec{r} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{r} \c...

Let a unit vector \(\widehat{O P}\) make angles \(\alpha, \beta, \gamma\) with the positive directions of the co-ordinate axes \(\mathrm{OX}\), \(\mathrm{OY}, \mathrm{OZ}\) respectively, where \(\beta \in\left(0, \frac{\pi}{2}\right)\). If ...

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) are three non-zero vectors and \(\widehat n\) is a unit vector perpendicular to \(\overrightarrow c\) such that $$\overrightarrow a = \alpha \overrightarrow b - \widehat n,(...

Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is :