Let `A ,B ,C` be points with position vectors `2 hat i- hat j+ hat k , hat i+2 hat j+ hat ka n d2 hat i+ hat j+2 hat k` respectively. Find the shortest distance between point `B` and plane `O A Cdot`

Show that the points A(-2 hat i+3 hat j+5 hat k), B( hat i+2 hat j+3 hat k) and C(7 hat i-3 hat k) are collinear.

Show that the point A , B and C with position vectors vec a =3 hat i - 4 hat j -4 hat k = 2 hat i j + hat k and vec c = hat i - 3 hat j - 5 hat k , respectively from the vertices of a right angled triangle.

Show that the points A,B,C with position vectors (3hat(i)- 2 hat(j)+ 4hat(k)),(hat(i) + hat(j) +hat(k)) and (-hat(i)+ 4 hat(j)- 2 hat(k)) respectively are collinear.

If the four points with positing vectors,-2hat i+hat j+hat k,hat i+hat j+hat k,hat j-hat k and lambdahat j+hat k are coplanar,then lambda is equal to

Find the image of the point with position vector (3hat i+hat j+2hat k) in the plane vec r*(2hat i-hat j+hat k)=4

Show that the four points A,B,C and D with position vectors 4hat i+5hat j+hat k,-(hat j+hat k),3hat j+9hat j+4hat k and 4(-hat i+hat j+hat k) respectively are coplanar.

vec A B=3 hat i- hat j+ hat ka n d vec C D=-3 hat i+2 hat j+4 hat k are two vectors. The position vectors of the points Aa n dC are =6 hat i+7 hat j+4 hat ka n d=-9 hat j+2 hat k respectively. Find the position vector of a point P on the line A B and a point Q on the line C D such that vec P Q is perpendicular to vec A Ba n d vec C B both.

Find the vector equation of the plane passing through three points with position vectors hat i+hat j-2hat k2hat i-hat j+hat k and hat i+2hat j+hat k Also find the coordinates of the point of intersection of this plane and the line vec r=3hat i-hat j-hat k+lambda(2hat i-2hat j+hat k)

Find the vector equation of the plane passing through three points with position vectors hat i+hat j-2hat k,hat i-hat j+hat k and hat i+2hat j+hat k .Also find the coordinates of the point of intersection of this plane and the line vec r=3hat i-hat j-hat k+lambda(2hat i-2hat j+hat k)

Find the angel between the vectors vec a=hat i-hat j+hat k and vec b=hat i+hat j-hat k