Find the vector equation of a line passing through `3 hat i-5 hat j+7 hat k` and perpendicular to theplane `3x-4y+5z=8.`

Find the vector equation of the line passing through (1, 2, 3 ) and parallel to the planes -> rdot( hat i- hat j+2 hat k)=5\ a n d\ -> rdot(3 hat i+ hat j+ hat k)=6.

If the vectors 3 hat i + lambda hat j + hat k and 2 hat i - hat j + 8 hat k are perpendicular, then lambda is

Find the vector equation of the line passing through (1,2,3) and parallel to the planes vec rdot( hat i- hat j+2 hat k)a n d vec rdot(3 hat i+ hat j+ hat k)=6.

Find the unit vector perpendicular to the plane vec rdot(2 hat i+ hat j+2 hat k)=5.

Prove that the vectors vec a = 3 hat i + hat j +3 hat k and vec b = hat i - hat k are perpendicular.

Prove that the vectors vec a= hat i+ 2 hat j +3 hat k and vec b=2 hat i- hat j are perpendicular.

Find the projection of vector hat i+3 hat j+7 hat k on the vector 7 hat i- hat j+8 hat kdot

Prove that the vectors 3 hat i + hat j + 3 hat k and hat i - hat k are perpendicular.

Find the equation of the plane passing through (3,4,-1), which is parallel to the plane vec rdot(2 hat i-3 hat j+5 hat k)+7=0.

Find the parametric form of vector eqution of the straight line passing through (-1,2,1) and paralle to the straight line vec(r)=(2hat(i)+3hat(j)-hat(k))+t(hat(i)-2hat(j)+hat(k)) and lines find the shortest distance between the lines.