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

Find the value of lambda , if the point with position vectors 3hat(i) - 2hat(j) - hat(k), 2hat(i) + 3hat(j) - 4hat(k), -hat(i) + hat(j) + 2hat(k) and 4hat(i) + 5hat(j) + lambda hat(k) are coplanar.

The value of p such that the vectors hat(i) + 3hat(j) - 2hat(k), 2hat(i) - hat(j) + 4hat(k) and 3hat(i) + 2hat(j) + p hat(k) are coplanar is

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

Find the projection of vector hat(i) + 3hat(j) + 7hat(k) on the vector 2hat(i)-3hat(j) + 6hat(k) .

Write the projection of the vector 7hat(i) + hat(j) - 4hat(k) on the vector 2hat(i) + 6hat(j) + 3hat(k) .

If the vectors 2hat(i) + 3hat(j) - 6hat(k) and 4hat(i) - m hat(j) - 12 hat(k) are parallel find m.

Find the magnitude of the vector (2hat(i) - 3hat(j) - 6hat(k)) + (-hat(i) + hat(j) + 4hat(k)) .

Find the sum of vectos a = hat(i)- 2hat(j)+hat(k),b=-2hat(i)+4hat(j)+5hat(k) and c=hat(i)-6hat(j)-7hat(k) .