Now $\vec{\beta}_2=\vec{\beta}-\vec{\beta}_1$
$=(\hat{ i }+2 \hat{ j }-4 \hat{ k })-\lambda(4 \hat{ i }+3 \hat{ j }+5 \hat{ k })$
$=(1-4 \lambda) \hat{ i }+(2-3 \lambda) \hat{ j }-(5 \lambda+4) \hat{ k }$
$\vec{\beta}_2 \cdot \vec{\alpha}=0$
$\Rightarrow 4(1-4 \lambda)+3(2-3 \lambda)-5(5 \lambda+4)=0$
$\Rightarrow 4-16 \alpha+6-9 \lambda-25 \lambda-20=0$
$\Rightarrow 50 \lambda=-10$
$\Rightarrow \lambda=\frac{-1}{5}$
$\vec{\beta}_2=\left(1+\frac{4}{5}\right) \hat{ i }+\left(2+\frac{3}{5}\right) \hat{ j }-(-1+4) \hat{ k }$
$\vec{\beta}_2=\frac{9}{5} \hat{ i }+\frac{13}{5} \hat{ j }-3 \hat{ k }$
$5 \vec{\beta}_2=9 \hat{ i }+13 \hat{ j }-15 \hat{ k }$
$5 \vec{\beta}_2 \cdot(\hat{ i }+\hat{ j }+\hat{ k })=9+13-15=7$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
વિધાન ${\text{ - 2 : }}$ રેખા $\frac{{\text{x}}}{{\text{1}}}\,\, = \,\,\frac{{y\,\, - \,\,1}}{2}\,\, = \,\,\frac{{z\,\, - \,\,2}}{3}\,$ એ $A\,\,\left( {1,\,\,0,\,\,7} \right)$ અને $B\,\,\left( {1,\,\,6,\,3} \right)$ ને જોડતા રેખાખડને લંબ-દ્રીભાજે છે