Evaluate `[[i, j, k]]`. Also, interpret it geometrically.

Show that the points, A, B and C having position vectors (2hat(i)-hat(j)+hat(k)), (hat(i)-3hat(j)-5hat(k)) and (3hat(i)-4hat(j)-4hat(k)) respectively are the vertices of a rightangled triangle. Also, find the remaining angles of the triangle.

Show that the points A, B, C with position vectors 2 hat i- hat j+ hat k ,\ hat i-3 hat j-5 hat k and 3 hat i-4 hat j-4 hat k respectively, are the vertices of a right angled triangle. Also, find the remaining angles of the triangle.

If veca=-vec(2i)-vec(2j)+vec(4k),vecb=-vec(2i)+vec(4j)-vec(2k) and vecc=vec(4i)-vec(2j)-vec(2k) Calculate the value of [veca vecb vecc\] and interpret the result.

If veca=-vec(2i)-vec(2j)+vec(4k),vecb=-vec(2i)+vec(4j)-vec(2k) and vecc=vec(4i)-vec(2j)-vec(2k) Calculate the value of [veca vecb vecc\] and interpret the result.

Find a unit vector perpendicular to each of the vectors hat i+2 hat j+3 hat k and -3 hat i-2 hat j+hat k . Also find the area of the parallelogram determined by these vectors.

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.

Interpret Lagrange's mean value theorem geometrically.

Find the foot of the perpendicular drawn from the point 2hat i-hat j+5hat k to the line vec r=(11hat i-2hat j-8hat k)+lambda(10hat i-4hat j-11hat k) Also find the length of the perpendicular.

Show that the points A,B,C with position vectors 2hat i-hat j+hat k,hat i-3hat j-5hat k and 3hat i-4hat j-4hat k respectively,are the vertices of a right angled triangle.Also,find the remaining angles of the triangle.