Vector Equations

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 $\left(a^2+b^2+c^2\right)$

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 $\left(1,2,3\right)$.

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

Express the vector $\hat{i} + 4\hat{j} -4\hat{k}$ as a linear combination of the vectors $2\hat{i} - \hat{j} + 3\hat{k} , \hat{i} - 2\hat{j} +4\hat{k}$ and $-\hat{i} +3\hat{j} -5\hat{k}$.

Express the vector $\overline{a} = 9\hat{i} + \widehat{j }+ 2\hat{k}$ as a linear combination of the vectors $\overline{q} = -\hat{i} - \hat{j} + 2\hat{k}$ and $\overline{r}=3\widehat{i }+\hat{j} -\hat{k}$.

Show that the points whose position vectors are $5\hat{i}+5\hat{k},-4\hat{i}+3\hat{j}-\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$ are collinear.

Show that the points whose position vectors are $5\hat{i}+5\hat{k}, 2\hat{i}+\hat{j}+3\hat{k}$ and $-4\hat{i}+3\hat{j}-\hat{k}$ are collinear.

The value of $\lambda$, if the vectors $\widehat{\dot{\mathbf{i}}}-2\widehat{\dot{\mathbf{j}}}+3\widehat{\mathbf{k}}, -2\widehat{\dot{\mathbf{i}}}+3\widehat{\dot{\mathbf{j}}}-4\widehat{\mathbf{k}}, \mathrm{\lambda }\mathrm{ }\widehat{\dot{\mathbf{i}}}-\widehat{\dot{\mathbf{j}}}+2\widehat{\mathbf{k}}$are linearly dependent is:

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