JEE Vector Algebra PYQs with Solutions PDF, Download Now

Let $$\overrightarrow{a}=-\widehat{i}+2\widehat{j}+2\widehat{k},\overrightarrow{b}=8\widehat{i}+7\widehat{j}-3\widehat{k} \text { and } \overrightarrow{c}$$ be a vector such that $$\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}$$. If $$\overrightarrow{c}\cdot(\widehat{i}+\widehat{j}+\widehat{k})=4$$, then $$\mid\overrightarrow{a}+\overrightarrow{c}\mid^{2}$$ is equal to :

Let $$\overrightarrow{c} \text{ and } \overrightarrow{d}$$ be vectors such that $$\mid\overrightarrow{c}+\overrightarrow{d}\mid=\sqrt{29}$$ and $$\overrightarrow{c}\times( 2\widehat{i}+3\widehat{j}+4\widehat{k})=(2\widehat{i}+3\widehat{j}+4\widehat{k})\times\overrightarrow{d}$$. If $$\lambda_{1}, \lambda_{2}( \lambda_{1}> \lambda_{2})$$ are the possible values of $$(\overrightarrow{c}+\overrightarrow{d})\cdot(-7\widehat{i}+2\widehat{j}+3\overrightarrow{k})$$, then the equation $$K^{2}x^{2}+(K^{2}-5K+\lambda_{1})xy+\left(3K+\frac{\lambda_{2}}{2} \right)y^{2}-8x+12y+\lambda_{2}=0$$ represents a circle, for K equal to :

Let $$(\alpha,\beta,\gamma)$$ be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line $$\overrightarrow{r}=(-\widehat{i}+3\widehat{j}+\widehat{k})+\lambda(2\widehat{i}+3\widehat{j}-\widehat{k}).$$ Then the length of the projection of the vector $$\alpha\widehat{i}+\beta\widehat{j}+\gamma\widehat{k}$$ on the vector $$6\widehat{i}+2\widehat{j}+3\widehat{k}$$ is:

Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}+\widehat{k}$$ and $$\overrightarrow{b}= \lambda \widehat{j}+2\widehat{k}, \lambda\in Z$$ be two vectors. Let $$\overrightarrow{c}= \overrightarrow{a} \times \overrightarrow{b} \text{and } \overrightarrow{d}$$ be a vector of magnitude 2 in yz-plane. If $$|\overrightarrow{c}|=\sqrt{53}$$, then the maximum possible value of $$\left(\overrightarrow{c}\cdot\overrightarrow{d}\right)^{2}$$ is equal to :

Let $$\overrightarrow{a}=-\widehat{i}+\widehat{j}+2\widehat{k},\overrightarrow{b}=\widehat{i}-\widehat{j}-3\widehat{k},\overrightarrow{c}=\overrightarrow{a} \times \overrightarrow{b}\text{ and }\overrightarrow{d}=\overrightarrow{c}\times\overrightarrow{a}$$. Then $$\large (\overrightarrow{a}-\overrightarrow{b}).\overrightarrow{d}$$ is equal to:

Let $$\overrightarrow{AB} = 2\widehat{i}+4\widehat{j}-5\widehat{k}$$ and $$ \overrightarrow{AD} = \widehat{i}+2\widehat{j}+\lambda\widehat{k}, \lambda\text{ }\epsilon \text{ } R$$. Let the projection of the vector $$ \overrightarrow{v}=\widehat{i}+\widehat{j}+\widehat{k}$$ on the disgonal $$\overrightarrow{AC}$$ of the parallelogram ABCD be of length one unit. If $$\alpha> \beta$$, be the roots of the equation $$\lambda^{2}x^{2}-6\lambda x+5=0$$, then $$2\alpha-\beta$$ is equal to

Let $$\overrightarrow{r}=2\widehat{i}+\widehat{j}-2\widehat{k}, \overrightarrow{b}=\widehat{i}+\widehat{j}\text{ and }\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$$. Let $$\overrightarrow{d}$$ be a vector such that $$|\overrightarrow{d}-\overrightarrow{a}|=\sqrt{11},|\overrightarrow{c}\times \overrightarrow{d}|=3$$ and the angle between $$\overrightarrow{c}\text{ and }\overrightarrow{d}$$ is $$\frac{\pi}{4}$$. Then $$\overrightarrow{a}. \overrightarrow{d}$$ is equal to

Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}-\widehat{k}, \overrightarrow{b}=\widehat{i}+ 3\widehat{j}-\widehat{k}$$ and $$\overrightarrow{c} = 2\widehat{i}+\widehat{j}+3\widehat{k}.$$ Let $$\overrightarrow{\nu}$$ be the vector in the plane of the vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, such that the length of its projection on the vector $$\overrightarrow{C}$$ is $$\frac{1}{\sqrt{14}}$$. Then $$\mid \overrightarrow{\nu} \mid$$ is euqal to

Let $$\overrightarrow{a}= 2\widehat{i}-5\widehat{j}+5\widehat{k}$$ and $$\overrightarrow{b}= \widehat{i}-\widehat{j}+3\widehat{k}$$. If $$\overrightarrow{C}$$ is a vector such that $$2(\overrightarrow{a}\times\overrightarrow{c})+3(\overrightarrow{b}\times\overrightarrow{c})= \overrightarrow{0}$$ and $$(\overrightarrow{a}-\overrightarrow{b})\cdot\overrightarrow{c}=-97,$$ then $$\mid \overrightarrow{c}\times\widehat{k} \mid^{2}$$ is equal to

For three unit vectors $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ satisfying $$|\overrightarrow{a}-\overrightarrow{b}|^{2}+|\overrightarrow{b}-\overrightarrow{c}|^{2}+|\overrightarrow{c}-\overrightarrow{a}|^{2}=9$$ and $$|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}|+3$$. the positive value of k is

Let PQR be a triangle such that $$\overrightarrow{PQ}=-2\widehat{i}-\widehat{j}+2\widehat{k}$$ and $$\overrightarrow{PR}=a\widehat{i}+b\widehat{j}-4\widehat{k},a,b \in Z$$. Let S be the point on QR, which is equidistant from the lines PQ and PR. If $$|\overrightarrow{PR}|=9$$ and $$\overrightarrow{PS}=\widehat{i}-7\widehat{j}+2\widehat{k}$$, then the value of 3a - 4b is_______

Let $$\overrightarrow{a}=\widehat{i}-2\widehat{j}+3\widehat{k}, \overrightarrow{b}=2\widehat{i}+\widehat{j}-\widehat{k}, \overrightarrow{c}=\lambda \widehat{i}+\widehat{j}+\widehat{k}$$ and $$\overrightarrow{v}= \overrightarrow{a} \times \overrightarrow{b}$$. If $$\overrightarrow{v}\cdot\overrightarrow{c}=11$$ and the length of the projection of $$\overrightarrow{b}$$ on $$\overrightarrow{c}$$ is p, then $$9p^{2}$$ is equal to

Let $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ be three vectors such that $$\overrightarrow{a}\times\overrightarrow{b}=2(\overrightarrow{a}\times\overrightarrow{c}).$$ If $$ \mid \overrightarrow{a}\mid, \mid\overrightarrow{b}\mid = 4, \mid \overrightarrow{c}\mid = 2,$$ and the angle between $$\overrightarrow{b}$$ and $$\overrightarrow{c}$$ is $$60^{o}$$, then $$\mid\overrightarrow{a}\cdot\overrightarrow{c}$$ is

Let $$\overrightarrow{a}$$ and $$ \overrightarrow{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{3}$$. Tf $$\lambda \overrightarrow{a} +2\overrightarrow{b}\text{ and }3\overrightarrow{a}-\lambda \overrightarrow{b}$$ are perpendicular to each other, then the number of values of $$\lambda$$ in [-1,3] is :

$$ \text{Let }\overrightarrow{c} \text{ be the projection vector of }\overrightarrow{b}=\lambda\widehat{i}+4\widehat{k}, \lambda > 0 , \text{ on the vector } \overrightarrow{a}=\widehat{i}+2\widehat{j}+2\widehat{k}. \text{ If } \mid \overrightarrow{a}+ \overrightarrow{c}\mid= 7, \text{ then the area of the parallelogram formed by the vectors }\overrightarrow{b} \text{ and }\overrightarrow{c} \text{ is } $$______

Let $$\vec a=\hat{i}+2\hat{j}+3\hat{k},  b=3\hat{i}+\hat{j}-\hat{k} $$ and  be three vectors such that  $$c$$ is coplanar with $$ \vec a$$  and  $$\vec b$$. If  $$\vec c $$ is perpendicular to  $$\vec b$$  and  $$\vec a\cdot \vec c=5,$$  then  $$|\vec c|$$  is equal to:

Let the position vectors of three vertices of a triangle be $$4\vec p+\vec q-3\vec r,\;-5\vec p+\vec q+2\vec r$$ and $$2\vec p-\vec q+2\vec r.$$ If the position vectors of the orthocenter and the circumcenter  of the triangle are $$\frac{\vec p+\vec q+\vec r}{4}$$ and $$\alpha\vec p+\beta\vec q+\gamma\vec r$$ { respectively, then $$\alpha+2\beta+5\gamma$$  is equal to:

$$\text{Let }\vec a=3\hat i-\hat j+2\hat k,\quad\vec b=\vec a\times(\hat i-2\hat k)\text{ and } \vec c=\vec b\times\hat k.\text{Then the projection of } (\vec c-2\hat j)\text{ on } \vec a \text{ is:}$$

Let $$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+2\hat{j}+\hat{k}$$ and $$\overrightarrow{d}=\overrightarrow{a}\times \overrightarrow{b}$$. If$$\overrightarrow{c}$$ is a vector such that $$\overrightarrow{a}. \overrightarrow{c}=|\overrightarrow{c}|,|\overrightarrow{c}-2\overrightarrow{a}|^{2}=8$$ and the angle between $$\overrightarrow{d}$$ and $$\overrightarrow{c}$$ is $$\frac{\pi}{4}$$, then $$|10-3\overrightarrow{b}.\overrightarrow{c}|+|\overrightarrow{d}\times \overrightarrow{c}|^{2}$$ is equal to

Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k}$$ and $$\overrightarrow{b}=2\hat{i}+7hat{j}+3\hat{k}$$. Let $$L_{1}:\overrightarrow{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \overrightarrow{a},\lambda \in R$$. and $$L_{2}: \overrightarrow{r}=(\hat{j}+\hat{k})+\mu \overrightarrow{b}, \mu \in R$$ be two lines. If the line $$L_{3}$$ passes through the point of intersection of $$L_{1}$$ and $$L_{2}$$, and is parallel to $$\overrightarrow{a}+\overrightarrow{b}$$, then $$L_{3}$$ passes through the point :

Let $$\overrightarrow{r}=2\hat{i}-\hat{j}+3\hat{k}, \overrightarrow{c}=3\hat{i}-5\hat{j}+\hat{k}$$ and $$\overrightarrow{c}$$ be a vector such that $$\overrightarrow{c} \times \overrightarrow{c} = \overrightarrow{c} \times \overrightarrow{b}$$ and $$(\overrightarrow{a}+\overrightarrow{c}).(\overrightarrow{b}.\overrightarrow{c})=168$$. Then the maximum value of $$|\overrightarrow{c}|^{2}$$ is :

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $$\widehat{i}+2\widehat{j}+\widehat{k},\widehat{i}+3\widehat{j}-2\widehat{k}$$ and $$2\widehat{i}+\widehat{j}-\widehat{k}$$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through of the triangle ABC at the point . If the length of AD is $$\frac{\sqrt{110}}{3}$$ and the volume of the tetrahedron is $$\frac{\sqrt{805}}{6\sqrt{2}}$$, then the position vector of is

Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that $$\frac{\text{lenght of arc AB}}{\text{lenght of arc BC}}=\frac{1}{5}$$,and $$\overrightarrow{OC}=\alpha\overrightarrow{OA}+\beta\overrightarrow{OB}$$, then $$\alpha +\sqrt{2}(\sqrt{3}-1)\beta$$ is equal to

Let $$\widehat{a}$$ be a unit vector perpendicular to the vectors $$\overrightarrow{b}=\widehat{i}-2\widehat{j}+3\widehat{k}$$ and $$\overrightarrow{c}=2\widehat{i}+3\widehat{j}-\widehat{k}$$, and makes an angle of $$\cos^{-1}(-\frac{1}{3})$$ with the vector $$\widehat{i}+\widehat{j}+\widehat{k}$$ . If $$\widehat{a}$$ makes an angle of $$\frac{\pi}{3}$$ with the vector $$\widehat{i}+\alpha\widehat{j}+\widehat{k}$$ , then the value of $$\alpha$$ is :

Let P be the foot of the perpendicular from the point (1,2,2) on the line L: $$\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}.$$ Let the line $$\vec{r}=(-\hat{i}+\hat{j}-2\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}), \quad \lambda \in \mathbb{R},$$ intersect the line L at Q. Then $$2(PQ)^{2}$$ is equal to:

Let a unit vector which makes an angle of 60° with $$2\hat{i} + 2\hat{j} - \hat{k}$$ and angle 45° with $$\hat{i} - \hat{k}$$ be $$\overrightarrow{C}$$. Then $$\overrightarrow{C} + \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)$$ is:

Let ABC be a triangle of area $$15\sqrt{2}$$ and the vectors $$\overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k}$$, $$\overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k}$$ and $$\overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k}$$, d > 0. Then the square of the length of the largest side of the triangle ABC is ______.

Let A, B, C be three points in $$xy-plane$$, whose position vector are given by $$\sqrt{3}\hat{i}+\hat{j}, \hat{i}+\sqrt{3}\hat{j}$$ and $$a\hat{i}+ (1-a)\hat{j}$$ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors $$\overrightarrow{OA}$$ and $$\overrightarrow{OB}$$ is $$\frac{9}{\sqrt{2}}$$, then the sum of all the possible values of $$a$$ is :

Let $$\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}$$, $$\vec{b} = \hat{i} + 2\hat{j} - 4\hat{k}$$ and $$\vec{c} = ((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$$. Then $$\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})$$ is equal to:

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k}$$ and $$\vec{c} = 4\hat{i} + c_2\hat{j} + c_3\hat{k}$$ be three vectors such that $$\vec{b} \times \vec{a} = \vec{c} \times \vec{a}$$. If the angle between the vector $$\vec{c}$$ and the vector $$3\hat{i} + 4\hat{j} + \hat{k}$$ is $$\theta$$, then the greatest integer less than or equal to $$\tan^2 \theta$$ is:

If $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 3(\hat{i} - \hat{j} + \hat{k})$$ and $$\vec{c}$$ be the vector such that $$\vec{a} \times \vec{c} = \vec{b}$$ and $$\vec{a} \cdot \vec{c} = 3$$, then $$\vec{a} \cdot ((\vec{c} \times \vec{b}) - \vec{b} - \vec{c})$$ is equal to

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 the position vectors of the vertices A, B and 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 $$l_3$$ be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides AB, BC and CA respectively, then $$l_1^2 + l_2^2 + l_3^2$$ equals :

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 $$\vec{a} + \alpha\vec{b} + \beta\vec{c} = \vec{0}$$, then $$\alpha + \beta$$ is equal to

Let $$O$$ be the origin and the position vector of $$A$$ and $$B$$ be $$2\hat{i} + 2\hat{j} + \hat{k}$$ and $$2\hat{i} + 4\hat{j} + 4\hat{k}$$ respectively. If the internal bisector of $$\angle AOB$$ meets the line $$AB$$ at $$C$$, then the length of $$OC$$ is

Let $$\vec{OA} = \vec{a}, \vec{OB} = 12\vec{a} + 4\vec{b}$$ and $$\vec{OC} = \vec{b}$$, where $$O$$ is the origin. If $$S$$ is the parallelogram with adjacent sides $$OA$$ and $$OC$$, then $$\frac{\text{area of the quadrilateral } OABC}{\text{area of } S}$$ is equal to _____

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 $$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k}, \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}}\hat{k}$$ and $$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$$ respectively. If $$\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}}\hat{k}$$, then $$|\hat{u} - \vec{v}|^2$$ is equal to

Let $$A(2, 3, 5)$$ and $$C(-3, 4, -2)$$ be opposite vertices of a parallelogram $$ABCD$$ if the diagonal $$\vec{BD} = \hat{i} + 2\hat{j} + 3\hat{k}$$ then the area of the parallelogram is equal to

Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$ and $$\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$$ be two vectors such that $$|\vec{a}| = 1$$; $$\vec{a} \cdot \vec{b} = 2$$ and $$|\vec{b}| = 4$$. If $$\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$$, then the angle between $$\vec{b}$$ and $$\vec{c}$$ is equal to :

Let $$\vec{a} = \hat{i} + \alpha\hat{j} + \beta\hat{k}$$, $$\alpha, \beta \in R$$. 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 $$\vec{a} \cdot \vec{b} = 3\sqrt{2}$$, then the value of $$(\alpha^2 + \beta^2)|\vec{a} \times \vec{b}|^2$$ is equal to

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 $$\vec{a} = 3\hat{i} + \hat{j} - 2\hat{k}$$, $$\vec{b} = 4\hat{i} + \hat{j} + 7\hat{k}$$ and $$\vec{c} = \hat{i} - 3\hat{j} + 4\hat{k}$$ be three vectors. If a vector $$\vec{p}$$ satisfies $$\vec{p} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{p} \cdot \vec{a} = 0$$, then $$\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k})$$ is equal to

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 $$192\sin^2\alpha$$ is equal to _________

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 $$(\vec{a} - \vec{b} + \hat{i}) \cdot \vec{c} = -3$$. Then $$|\vec{c}|^2$$ is equal to

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 $$\vec{a} \cdot \vec{d} = 1$$, then $$(\vec{a} \times \vec{b}) \cdot \vec{c}$$ is equal to

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 perpendicular, then the value of $$(14 \cos \theta)^2$$ is equal to

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 the minimum value of $$27|\vec{c} - \vec{a}|^2$$ is equal to:

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 $$\vec{a} \cdot \vec{c} = -29$$, then $$\vec{c} \cdot (-2\hat{i} + \hat{j} + \hat{k})$$ is equal to:

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

Let $$\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \vec{b} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a})$$. If $$\vec{a} \cdot \vec{c} = 130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to ______