In a triangle ABC, the sides AB and AC represented by the vector `3hat(i)+hat(j)+hat(k)and hat(i)+2hat(j)+hat(k)` respectively. Calcutate the angle `angleABC`.

In a triangle ABC, the sides AB and BC are represented by vectors 2hat(i) - hat(j) + 2hat(k), hat(i) + 3hat(j) + 5hat(k) respectively. Find the vector representing CA.

The adjacent sides AB and AC of a triangle ABC are represented by the vectors -2hat(i)+ 3hat(j) + 2hat(k) and -4hat(i) + 5hat(j) + 2hat(k) respectively. The area of the triangle ABC is

The adjacent sides AB and AC of a triangle ABC are represented by the vectors -2 hat(i) + 3 hat(j) + 2 hat(k) and -4hat(i) + 5 hat(j) + 2 hat(k) respectively. The area of the triangle ABC is

The area of the parallelogram whose sides are represented by the vector hat(j)+3hat(k) and hat(i)+2hat(j)-hat(k) is

Find the angle between the vectors 2 hat(i) - hat(j) - hat(k) and 3 hat(i) + 4 hat(j) - hat(k) .

(i) Determine the value of p for which the vectors phat(i)-5hat(j) " and " 2hat(i)-3hat(j) are collinear. (ii) The sides AB and BC of the triangle ABC are represented by the vectors 2hat(i)-hat(j)+2hat(k) " and " hat(i)+3hat(j)+5hat(k) respectively, find the vector representing side CA of the triangle ABC.

Show that the vectors 2hat(i)-hat(j)+hat(k) and hat(i)-3hat(j)-5hat(k) are at right angles.