Vector Algebra

Let $\lambda \in \mathbb{R}, \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}$.
If $((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}$, then $|\l...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three non-zero non-coplanar vectors. Let the position vectors of four points \(A,B,C\) and \(D\) be $$\overrightarrow a - \overrightarrow b + \overrightarrow...

If the vectors \(\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k\), \(\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k\) and \(\overrightarrow c = 2\widehat i + 3\widehat j + \widehat k\) are coplanar ...

Let \(\overrightarrow a = 4\widehat i + 3\widehat j\) and \(\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k\). If \(\overrightarrow c\) is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightar...

If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \tim...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three non zero vectors such that \(\overrightarrow b\) . \(\overrightarrow c\) = 0 and $$\overrightarrow a \times (\overrightarrow b \times \overrightarrow...

The vector \(\overrightarrow a = - \widehat i + 2\widehat j + \widehat k\) is rotated through a right angle, passing through the y-axis in its way and the resulting vector is \(\overrightarrow b\). Then the projection of $$3\overrightarr...

If the four points, whose position vectors are \(3\widehat i - 4\widehat j + 2\widehat k,\widehat i + 2\widehat j - \widehat k, - 2\widehat i - \widehat j + 3\widehat k\) and \(5\widehat i - 2\alpha \widehat j + 4\widehat k\) are coplanar, ...

Let \(\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1\) and \(\overrightarrow a \times \overrightarrow b = \widehat i - \widehat j\). Then $$\overrightarrow a - 6\overrightarrow ...

Let \(\overrightarrow u = \widehat i - \widehat j - 2\widehat k,\overrightarrow v = 2\widehat i + \widehat j - \widehat k,\overrightarrow v .\,\overrightarrow w = 2\) and $$\overrightarrow v \times \overrightarrow w = \overrightarrow u...

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that \({{QA} \over {AR}} = {{RB} \over {BP}} = {{PC} \over {CQ}} = {1 \over 2}\). Then \({{Area(\Delta PQR)} \over {Area(\Delta ABC)}}\) is equal ...

Let $$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrightarrow a \,.\,\overrightarrow c = 7,2\overrightarrow b \,.\,\overrightarrow c + 43 = 0,\...

Let \(\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k\) and \(\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k\). Let \({\overrightarrow \beta _1}\) be parallel to \(\overrightarrow \alpha\) and $${\ove...

\(A(2,6,2), B(-4,0, \lambda), C(2,3,-1)\) and \(D(4,5,0),|\lambda| \leq 5\) are the vertices of a quadrilateral \(A B C D\). If its area is 18 square units, then \(5-6 \lambda\) is equal to __________.

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\) and \(\vec{u}\) be a vector such that \(|\vec{u}|=\alpha>0\). If the minimum value of the scalar triple product $$\left[ {\matrix{
{\overrightar...

Let \(\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}\) be three given vectors. If \(\overrightarrow{\mathrm{r}}\) is a vector such that $$\vec{r} \times \vec{a}=\vec{c} \times \vec...

Let \(\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}\) be two vectors. Then which one of the following statements is TRUE ?

Let $\mathrm{ABCD}$ be a quadrilateral. If $\mathrm{E}$ and $\mathrm{F}$ are the mid points of the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ respectively and $(\overrightarrow{A B}-\overrightarrow{B C})+(\overrightarrow{A D}-\overrightarrow...

Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k}, \hat{i}+2 \hat{j}+\mu \hat{k}$ and $3 \hat{i}-4 \hat{j}+5 \hat{k}$, where $\lambda-\mu=5$, are coplanar, then $\sum\limits_{(\lambda, \mu) \i...

Let \(\vec{a}=3 \hat{i}+\hat{j}-\hat{k}\) and \(\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}\). If \(\vec{b}\) is a vector such that \(\vec{a}=\vec{b} \times \vec{c}\) and \(|\vec{b}|^{2}=50\), then \(|72-| \vec{b}+\left.\vec{c}\right|^{2} \mid\) ...

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

Let for a triangle \(\mathrm{ABC}\),
\(\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}\)
\(\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}\)
$$\overrightarrow{\mathrm{CA}}=4 \hat{i}+3 \hat{j}+\delta \hat{k...

Let \(|\vec{a}|=2,|\vec{b}|=3\) and the angle between the vectors \(\vec{a}\) and \(\vec{b}\) be \(\frac{\pi}{4}\). Then \(|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^{2}\) is equal to :

Let \(\lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{c}\) be a vector such that $$(\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=...

Let \(a, b, c\) be three distinct real numbers, none equal to one. If the vectors \(a \hat{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{j}+\hat{\mathrm{k}}\) and \(\hat{\mathrm{i}}+\hat{\mathrm{j}}+c \hat{\mathrm{k}}\) are ...

Let \(\vec{a}\) be a non-zero vector parallel to the line of intersection of the two planes described by \(\hat{i}+\hat{j}, \hat{i}+\hat{k}\) and \(\hat{i}-\hat{j}, \hat{j}-\hat{k}\). If \(\theta\) is the angle between the vector $$\vec{a}$...

For any vector \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}\), with \(10\left|a_{i}\right|<1, i=1,2,3\), consider the following statements :
(A): $$\max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\} \leq|\ve...

Let \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\). If \(\vec{c}\) is a vector such that \(\vec{a} \cdot \vec{c}=11, \vec{b} \cdot(\vec{a} \times \vec{c})=27\) and $$\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$...

If four distinct points with position vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) are coplanar, then \([\vec{a} \,\,\vec{b} \,\,\vec{c}]\) is equal to :

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If \(\overrightarrow {OP} = \overrightarrow u ,\overrightarrow {OR} = \overrightarrow v\), and $$\overrightarrow {OQ} = \alpha \overrightarr...

Let O be the origin and the position vector of the point P be \(- \widehat i - 2\widehat j + 3\widehat k\). If the position vectors of the points A, B and C are $$ - 2\widehat i + \widehat j - 3\widehat k,2\widehat i + 4\widehat j - 2\wide...

If the points \(\mathrm{P}\) and \(\mathrm{Q}\) are respectively the circumcenter and the orthocentre of a \(\triangle \mathrm{ABC}\), then \(\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}\) is
equal ...

Let \(\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{d}\) be a vector which is perpendicular to both \(\vec{a}\) and \(\vec{b}\), and \(\vec{c} \cdot \vec{d}=12\). Th...

Let a vector \(\overrightarrow c\) be coplanar with the vectors \(\overrightarrow a = - \widehat i + \widehat j + \widehat k\) and \(\overrightarrow b = 2\widehat i + \widehat j - \widehat k\). If the vector \(\overrightarrow c\) also ...

Let \(\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k\), \(\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k\) and \(\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k\) where $$\alpha ,\,\beta \i...

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

Let A, B, C be three points whose position vectors respectively are
\(\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k\)
\(\overrightarrow b = 2\widehat i + \alpha \widehat j + 4\widehat k,\,\alpha \in R\)
$$\overrightarrow c ...

Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{4}\). If \(\theta\) is the angle between the vectors \((\hat{a}+\hat{b})\) and \((\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))\), then the ...

Let \(\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}\) and \(\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}\). Let \(\overrightarrow{\mathrm{c}}\) be a vector satisfying $$\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}}...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3\) and \(|\vec{a} \times \vec{b}|^{2}=75\). Then \(|\vec{a}|^{2}\) is equal to __________.

Let \(\vec{a}, \vec{b}, \vec{c}\) be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and
$$(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})+(\vec{b} \times \...

If \(\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k\), \(\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k\) and \(\overrightarrow c = {c_1}\widehat i + {c_2}\widehat j + {c_3}\widehat k\) are coplanar vectors and $$...

Let \(\overrightarrow a\) be a vector which is perpendicular to the vector \(3\widehat i + {1 \over 2}\widehat j + 2\widehat k\). If $$\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\wid...

Let \(\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k\) and \(\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k\), where \(\alpha \in R\). If the area of the parallelogram whose adjacent sides are repre...

Let a vector \(\vec{a}\) has magnitude 9. Let a vector \(\vec{b}\) be such that for every \((x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}\), the vector \((x \vec{a}+y \vec{b})\) is perpendicular to the vector $$(6 y \vec{a}-18 x \vec{b}...

Let the vectors \(\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}\) and \(\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R}\) be such that for $$\alpha, \beta, \gamma \in \mathbf{R}, \alpha \ve...

Let S be the set of all a \(\in R\) for which the angle between the vectors \(\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}\) and \(\vec{v}=\left(\log _{e} b\right) \hat{i}+2 \hat{j}+2 a\left(\log _{e} b\right) \hat{k}\), $...

Let \(\overrightarrow a = \widehat i + \widehat j - \widehat k\) and \(\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k\). Then the number of vectors \(\overrightarrow b\) such that $$\overrightarrow b \times \overrightarrow ...

Let \(\overrightarrow a\) and \(\overrightarrow b\) be the vectors along the diagonals of a parallelogram having area \(2\sqrt 2\). Let the angle between \(\overrightarrow a\) and \(\overrightarrow b\) be acute, $$|\overrightarrow a | ...

\(\text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2} \text {, then }|\vec{b} \times 2 \hat{j}|\)...