Linearly Dependent and Independent Vectors

If $\overrightarrow{x},\overrightarrow{y}$ are two non-zero and non-collinear vectors satisfying $\left[\left(a-2\right){\alpha }^2+\left(b-3\right)\alpha +c\right]\overrightarrow{x}+\left[\left(a-2\right){\beta }^2+\left(b-3\right)\beta +c\right]\overrightarrow{y}+\left[\left(a-2\right){\gamma }^2+\left(b-3\right)\gamma +c\right]\left(\overrightarrow{x}\times \overrightarrow{y}\right)=\mathrm{0\; }$

where $\alpha ,\beta ,\gamma$ are three distinct real numbers, then find the value of $\left(a^2+b^2+c^2\right)$

If $x\overrightarrow{a}+y\overrightarrow{b}+z\overrightarrow{c}=\overrightarrow{0}$, where $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar, then $\left(x,y,z\right)=$

If two of the three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are units vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ and $2\left(\overrightarrow{a}·\overrightarrow{b}+\overrightarrow{b}·\overrightarrow{c}+\overrightarrow{c}·\overrightarrow{a}\right)+3=0$, then the length of the third vector is

If the vectors $\overrightarrow{a}+1343\overrightarrow{b}+\overrightarrow{c}, -\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ and $\overrightarrow{a}+\mu \overrightarrow{b}+2\overrightarrow{c}$ are linearly dependent then $\mu =$

The vectors $2\overline{i}-3\overline{j}+4\overline{k}, \overline{i}-2\overline{j}+3\overline{k}, 3\overline{i}+\overline{j}-2\overline{k}$ are

Let $\overrightarrow{a}=\hat{j}-\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}-\hat{k}$. Then the vector $\overrightarrow{b}$ satisfying $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}=0$ and $\overrightarrow{a}·\overrightarrow{b}=3$ is

State fundamental theorem of space curves. Find the distance of plane $2x+3y-4=0$ from point $\left(1,2,3\right)$.

State fundamental theorem of space curves. Find the distance of plane $3x+2y-4=0$ from point $\left(1,2,3\right)$.

State fundamental theorem of space curves. Find the distance of plane $2z+x-2=0$ from point $\left(1,2,3\right)$.

State fundamental theorem of space curves. Find the distance of plane $4x+2z=1$ from point $\left(1,2,3\right)$.

State fundamental theorem of plane curves. Find the distance of plane $2x+3y-4=0$ from point $\left(1,2,3\right)$.

State fundamental theorem of plane curves. Find the distance of plane $3x+2y-4=0$ from point $\left(1,2,3\right)$.

State fundamental theorem of plane curves. Find the distance of plane $2z+x=2$ from point $\left(1,2,3\right)$.

State fundamental theorem of plane curves. Find the distance of plane $4x+2z=1$ from point $\left(1,2,3\right)$.

State fundamental theorem of plane curves. Find the distance of plane $2x+3z=4$ from point $\left(1,2,3\right)$.

State fundamental theorem of space curves. Find the distance of plane $2x+3z=4$ from point $\left(1,2,3\right)$.

Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be non-collinear vectors in $R^2$. Let $\overrightarrow{w}$ be the orthogonal projection vector of $\overrightarrow{u}$ on $\overrightarrow{v}$. Consider two statements:
(i) Any vector in $R^2$ can be written as a linear combination of $\overrightarrow{u}$ and $\overrightarrow{v}$
(ii) $\overrightarrow{w}$ can be written as a linear combination of $\overrightarrow{u}$ and $\overrightarrow{v}$ as $\overrightarrow{w}=a \overrightarrow{u}+b \overrightarrow{v}$ where both $a$and $b$ are nonzero real numbers.

If the vectors $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-collinear vectors and a triangle $\mathrm{ABC}$ with side lengths $\mathrm{a}, \mathrm{b}, \mathrm{c}$ satisfying the equation 

$(20a-15b)\overrightarrow{x}+(15b-12c)\overrightarrow{y}+(12c-20a)(\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0}$ Then the triangle $\mathrm{ABC}$ is 

Let $PQRS$ be a quadrilateral. If $M$ and $N$ are midpoints of the sides $PQ$ and $RS$ respectively then $\overrightarrow{PS}+\overrightarrow{QR}=$

Four vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ and $\overrightarrow{x}$ satisfy the relation $\left(\overrightarrow{a}·\overrightarrow{x}\right)\overrightarrow{b}=\overrightarrow{c}+\overrightarrow{x}$ where $\overrightarrow{b}·\overrightarrow{a}\ne 1$. The value of $\overrightarrow{x}$ in terms of $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ will be equal to