Find the vector equation of the plane passing through the points having position vectors ` hat i+ hat j-2 hat k ,2 i - hat j+ hat ka n d hat i+2 hat j+ hat kdot`

Find the area of triangle whose vertices have position vectors hat (i) + hat (j) + 2 hat (k) , 2 hat (i) + 2 hat (j) + 3 hat (k) and 3 hat (i) - hat (j) - hat (k)

Find the equation of the sphere described on the joint of points Aa n dB having position vectors 2 hat i+6 hat j-7 hat ka n d-2 hat i+4 hat j-3 hat k , respectively, as the diameter. Find the center and the radius of the sphere.

Find the value of lambda for which the three points with position vector 3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k , and4 hat i+5 hat j+lambda hat k are coplanar.

Find the vector equation of a line which passes through the point with position vector hat(i)-2hat(j)+4hat(k) and is in the direction of hat(i)+2hat(j)-hat(k) . Also, reduce it to cartesian from.

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

(i) Find the vector equation of a line passing through a point with position vector 2hat(i)-hat(j)+hat(k) , and parallel to the line joining the points -hat(i)+4hat(j)+hat(k) and hat(i)+2hat(j)+2hat(k) . Also find the cartesian equivalent of this equation. (ii) The cartesian equations of a line are 6x-2=3y+1=2z-2 . Find its direction ratios and also find vector equation of the line.

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

Find the angle between the vectors hat i-2 hat j+3 hat k and 3 hat i-2 hat j+ hat kdot

Prove that point hat i +2 hat j - 3 hat k ,2 hat i - hat j + hat k and 2 hat i + 5 hat j - hat k from a triangle in space.

Check whether the three vectors 2 hat i+2 hat j+3 hat k ,-3 hat i+3 hat j+2 hat ka n d3 hat i+4 hat k from a triangle or not