Two adjacent sides of a parallelogram `A B C D` are `2 hat i+4 hat j-5 hat k` and ` hat i+2 hat j+3 hat k` . Then the value of `|A CxxB D|` is a. `20sqrt(5)` b. `22sqrt(5)` c. `24sqrt(5)` d. `26sqrt(5)`

The area of the parallelogram whose adjacent sides are hat(i) + hat(k) and 2 hat(i) + hat(j) + hat(k) is ?

Show that the area of a parallelogram having diagonals 3 hat i+ hat j-2 hat k\ a n d\ hat i-3 hat j+4 hat k is 5sqrt(3) square units.

The sides of a parallelogram are 2 hat i+4 hat j-5 hat k and hat i+2 hat j+3 hat k . The unit vector parallel to one of the diagonals is a. 1/7(3 hat i+6 hat j-2 hat k) b. 1/7(3 hat i-6 hat j-2 hat k) c. 1/(sqrt(69))( hat i+6 hat j+8 hat k) d. 1/(sqrt(69))(- hat i-2 hat j+8 hat k)

Two adjacent sides of a parallelogram A B C D are given by vec A B=2 hat i+10 hat j+11 hat ka n d vec A D=- hat i+2 hat j+2 hat kdot The side A D is rotated by an acute angle alpha in the plane of the parallelogram so that A D becomes A D^(prime)dot If A D ' makes a right angle with the side A B , then the cosine of the angel alpha is given by a. 8/9 b. (sqrt(17))/9 c. 1/9 d. (4sqrt(5))/9

Two adjacent sides of a parallelogram A B C D are given by vec A B=2 hat i+10 hat j+11 hat ka n d vec A D=- hat i+2 hat j+2 hat kdot The side A D is rotated by an acute angle alpha in the plane of the parallelogram so that A D becomes A D^(prime)dot If A D ' makes a right angle with the side A B , then the cosine of the angel alpha is given by 8/9 b. (sqrt(17))/9 c. 1/9 d. (4sqrt(5))/9

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.

If the position vectors of P\ a n d\ Q are hat i+3 hat j-7 hat k\ a n d\ 5 hat i-2 hat j+4 hat k them the cosine of the angle between vec P Q and y-axis is 5/(sqrt(162)) b. 4/(sqrt(162)) c. -5/(sqrt(162)) d. (11)/(sqrt(162))

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- hat k) are collinear.

If theta is the angle between the vectors 2 hat i-2 hat j+4 hat k\ a n d\ 3 hat i+ hat j+2 hat k , then sintheta= 2/3 b. 2/(sqrt(7)) c. (sqrt(2))/7 d. sqrt(2/7)

If the scalar projection of vector x hat i- hat j+ hat k on vector 2 hat i- hat j+5 hat k , is 1/(sqrt(30)) ,then find the value of xdot