Let `x^2+3y^2=3` be the equation of an ellipse in the `x-y` plane. `Aa n dB` are two points whose position vectors are `-sqrt(3) hat ia n d-sqrt(3) hat i+2 hat kdot` Then the position vector of a point `P` on the ellipse such that `/_A P B=pi//4` is a. `+- hat j` b. `+-( hat i+ hat j)` c. `+- hat i` d. none of these

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

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

Show that the poins whose position vectors are 4hat(i)+5hat(k),hat(i)+hat(j)+3hat(k)" and "-5hat(i)+3hat(j)-hat(k) are collinear.

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.

A line passes through the point with position vector 2 hat i-3 hat j+4 hat k and is in the direction of 3 hat i+4 hat j-5 hat kdot Find the equations of the line in vector and Cartesian forms.

The vector(s) which is/are coplanar with vectors hat i+ hat j+2 hat k and hat i+2 hat j+ hat k , and perpendicular to vector hat i+ hat j+ hat k , is/are a. hat j- hat k b. - hat i+ hat j c. hat i- hat j d. - hat j+ hat k

If 4 hat i+7 hat j+8 hat k ,2 hat i+3 hat j+4hat ja n d2 hat i+5 hat j+7 hat k are the position vectors of the vertices A ,Ba n dC , respectively, of triangle A B C , then the position vecrtor of the point where the bisector of angle A meets B C is a. 2/3(-6 hat i-8 hat j- hat k) b. 2/3(6 hat i+8 hat j+6 hat k) c. 1/3(6 hat i+13 hat j+18 hat k) d. 1/3(5 hat j+12 hat k)

Find the resultant of vectors vec a= hat i- hat j+2 hat k and vec b= hat i+2 hat j-4 hat kdot Find the unit vector in the direction of the resultant vector.

The ratio in which the line segment joining the points whose position vectors are 2 hat i-4 hat j-7 hat ka n d-3 hat i+5 hat j-8 hat k is divided by the plane whose equation is hat r.( hat i-2 hat j+3 hat k)=13 is a. 13 : 12 internally b. 12 : 25 externally c. 13 : 25 internally d. 37 : 25 internally