Question

Find vector equation of plane which passes through the points (3,2,1) and (0,1,7) and is parallel to line r=2i-j+k+l(i-j-k)

Answer 1

$\text{The given line is parallel to the vector }\overrightarrow{i}-\overrightarrow{j}-\overrightarrow{k}$

$\text{Hence the required plane is passes through (3,2,1) and (0,1,7) }$

$\text{and parallel to the vector }\;\overrightarrow{i}-\overrightarrow{j}-\overrightarrow{k}$

$\text{The equation of the required plane is}$

$\left|\begin{array}{ccc}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\l&m&n\end{array}\right|=0$

$\left|\begin{array}{ccc}x-3&y-2&z-1\\0-3&1-2&7-1\\1&-1&-1\end{array}\right|=0$

$\left|\begin{array}{ccc}x-3&y-2&z-1\\-3&-1&6\\1&-1&-1\end{array}\right|=0$

$\text{Expanding along the first row, we get}$

$(x-3)(-1+6)-(y-2)(-3-6)+(z-1)(3+1)=0$

$5(x-3)+9(y-2)+4(z-1)=0$

$5x-15+9y-18+4z-4=0$

$\implies\bf\,5x+9y+4z-37=0$