Vector Algebra

Let $\vec a$ and $\vec b$ be the vectors along the diagonals of a parallelogram having area $2\sqrt2$. Let the angle between $\vec a$ and $\vec b$ be acute, $|\vec a|=1$, and $\left|\vec a\cdot\vec b\right|=\left|\vec a\times\vec b\right|$. If $\vec c=2\sqrt2\left(\vec a\times\vec b\right)-2\vec b$, then an angle between $\vec b$ and $\vec c$ is

If $\vec a=2\hat i+\hat j+3\hat k$, $\vec b=3\hat i+3\hat j+\hat k$ and $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$ are coplanar vectors and $\vec a\cdot\vec c=5$, $\vec b\perp\vec c$, then $122\left(c_1+c_2+c_3\right)$ is equal to ______.

Let $\vec a=\alpha\hat i+2\hat j-\hat k$ and $\vec b=-2\hat i+\alpha\hat j+\hat k$, where $\alpha\in\mathbf R$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec a$ and $\vec b$ is $\sqrt{15\left(\alpha^2+4\right)}$, then the value of $2\left|\vec a\right|^2+\left(\vec a\cdot\vec b\right)\left|\vec b\right|^2$ is equal to :

Let the plane $ax+by+cz=d$ pass through $(2,3,-5)$ and is perpendicular to the planes $2x+y-5z=10$ and $3x+5y-7z=12$.

If $a,b,c,d$ are integers $d>0$ and $\gcd(|a|,|b|,|c|,d)=1$, then the value of $a+7b+c+20d$ is equal to :

Let $\vec a$ be a vector which is perpendicular to the vector $3\hat i+\dfrac12\hat j+2\hat k$. If $\vec a\times\left(2\hat i+\hat k\right)=2\hat i-13\hat j-4\hat k$, then the projection of the vector $\vec a$ on the vector $2\hat i+2\hat j+\hat k$ is :

Let $\vec a=\alpha\hat i+3\hat j-\hat k$, $\vec b=3\hat i-\beta\hat j+4\hat k$ and $\vec c=\hat i+2\hat j-2\hat k$ where $\alpha,\beta\in\mathbf R$, be three vectors. If the projection of $\vec a$ on $\vec c$ is $\dfrac{10}3$ and $\vec b\times\vec c=-6\hat i+10\hat j+7\hat k$, then the value of $\alpha+\beta$ is equal to :

Let $P_1:\vec r\cdot\left(2\hat i+\hat j-3\hat k\right)=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)$, $(2,-2,-3)$ and $(1,-4,2)$. If the direction ratios of the line of intersection of $P_1$ and $P_2$ be $16,\alpha,\beta$, then the value of $\alpha+\beta$ is equal to ______.

Let A, B, C be three points whose position vectors respectively are
$$\vec a=\hat i+4\hat j+3\hat k$$
$$\vec b=2\hat i+\alpha\hat j+4\hat k,\ \alpha\in\mathbb R$$
$$\vec c=3\hat i-2\hat j+5\hat k$$
If $\alpha$ is the smallest positive integer for which $\vec a$, $\vec b$, $\vec c$ are noncollinear, then the length of the median, in $\Delta ABC$, through A is :

Let $\vec a=\hat i-2\hat j+3\hat k$, $\vec b=\hat i+\hat j+\hat k$ and $\vec c$ be a vector such that $\vec a+\left(\vec b\times\vec c\right)=\vec0$ and $\vec b\cdot\vec c=5$. Then, the value of $3\left(\vec c\cdot\vec a\right)$ is equal to ______.

Let a vector $\vec c$ be coplanar with the vectors $\vec a=-\hat i+\hat j+\hat k$ and $\vec b=2\hat i+\hat j-\hat k$. If the vector $\vec c$ also satisfies the conditions $\vec c\cdot\left[\left(\vec a+\vec b\right)\times\left(\vec a\times\vec b\right)\right]=-42$ and $\left(\vec c\times\left(\vec a-\vec b\right)\right)\cdot\hat k=3$, then the value of $\left|\vec c\right|^2$ is equal to :

Let ABC be a triangle such that $\overrightarrow{BC}=\vec a$, $\overrightarrow{CA}=\vec b$, $\overrightarrow{AB}=\vec c$, $\left|\vec a\right|=6\sqrt2$, $\left|\vec b\right|=2\sqrt3$ and $\vec b\cdot\vec c=12$. Consider the statements :

(S1) : $\left|\left(\vec a\times\vec b\right)+\left(\vec c\times\vec b\right)\right|-\left|\vec c\right|=6\left(2\sqrt2-1\right)$

(S2) : $\angle ACB=\cos^{-1}\left(\sqrt{\dfrac23}\right)$

Then

Let $\vec a=\hat i-\hat j+2\hat k$ and let $\vec b$ be a vector such that $\vec a\times\vec b=2\hat i-\hat k$ and $\vec a\cdot\vec b=3$. Then the projection of $\vec b$ on the vector $\vec a-\vec b$ is :

Let $\vec a=\alpha\hat i+\hat j-\hat k$ and $\vec b=2\hat i+\hat j-\alpha\hat k$, $\alpha>0$. If the projection of $\vec a\times\vec b$ on the vector $-\hat i+2\hat j-2\hat k$ is 30, then $\alpha$ is equal to :

A vector $\vec a$ is parallel to the line of intersection of the plane determined by the vectors $\hat i$, $\hat i+\hat j$ and the plane determined by the vectors $\hat i-\hat j$, $\hat i+\hat k$. The obtuse angle between $\vec a$ and the vector $\vec b=\hat i-2\hat j+2\hat k$ is

Let $\vec a=\alpha\hat i+\hat j+\beta\hat k$ and $\vec b=3\hat i-5\hat j+4\hat k$ be two vectors, such that $\vec a\times\vec b=-\hat i+9\hat j+12\hat k$. Then the projection of $\vec b-2\vec a$ on $\vec b+\vec a$ is equal to

Let $\vec a=2\hat i-\hat j+5\hat k$ and $\vec b=\alpha\hat i+\beta\hat j+2\hat k$. If $\left(\left(\vec a\times\vec b\right)\times\hat i\right)\cdot\hat k=\dfrac{23}2$, then $\left|\vec b\times2\hat j\right|$ is equal to

Let $\vec a,\vec b,\vec c$ be three non-coplanar vectors such that $\vec a\times\vec b=4\vec c$, $\vec b\times\vec c=9\vec a$ and $\vec c\times\vec a=\alpha\vec b$, $\alpha>0$. If $\left|\vec a\right|+\left|\vec b\right|+\left|\vec c\right|=\dfrac1{36}$, then $\alpha$ is equal to ______.

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 $\alpha,\beta,\gamma\in\mathbf R$, $\alpha\vec a+\beta\vec b+\gamma\vec c=\vec0\Rightarrow\alpha=\beta=\gamma=0$. Then, the set of all values of t is :

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 $\left(x\vec a+y\vec b\right)$ is perpendicular to the vector $\left(6y\vec a-18x\vec b\right)$. Then the value of $\left|\vec a\times\vec b\right|$ is equal to :

Let $P(-2,-1,1)$ and $Q\left(\dfrac{56}{17},\dfrac{43}{17},\dfrac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2+\beta^2$ is equal to ______.