In each of the following show that the given vectors are coplanar:
` 4 hat (i) + 2 hat (j) + hat (k) , 2 hat (i) - hat (j) + 3 hat (k) , 8 hat (i) + 7 hat (k) `

In each of the following show that the given vectors are coplanar: vec(a) = hat (i) + hat (j) - 6 hat (k) , vec(b) = hat (i) + 3 hat (j) + 4 hat (k) , vec(c) = 2 hat (i) + 5 hat (j) + 3 hat (k)

Show that the vectors are mutually perpendicular hat (i) + 2 hat (j) + hat (k) ,hat (i) + hat (j) - 3 hat (k) and 7 hat (i) - 4 hat (j) + hat (k)

The position vectors of four points A,B,C and D are given below . In each case , using vector method prove that the four points A,B,C and D are coplanar . 4 hat (i) + 8 hat (j) + 12 hat (k) , 2 hat (i) + 4 hat (j) + 6 hat(k), 3 hat (i) + 5 hat (j) + 4 hat(k) , 5 hat (i) + 8hat (j) + 5 hat (k)

Find the scalar and vector projections of 3 hat (i) - hat (j) + 4 hat (k) on 2 hat (i) + 3 hat (j) - 6 hat (k)

In each of the following cases two vectors are perpendicular to each other, find m m hat (i) - 2 hat (j) + hat(k) and 3 hat (i) - 2 hat (j) - 7 hat (k)

Find the area of triangle whose vertices have position vectors hat (i) + hat (j) + 2 hat (k) , 2 hat (i) + 2 hat (j) + 3 hat (k) and 3 hat (i) - hat (j) - hat (k)

If the vectors ahat (i) + a hat (j) + c hat (k) , hat (i) + hat (k) and c hat (i) + c hat (j) + b hat (k) be coplanar, show that c^(2) = ab

Check whether the given three vectors are coplanar or non-coplanar. -2 hat i-2 hat j+4 hat k ,-2 hat i+4 hat j ,4 hat i-2 hat j-2 hat k

Find the area of triangle (i) drawn on the vectors vec(a) = 6 hat (i) + 2 hat (j) - 3 hat (k) and vec (b) = 4 hat (i) - hat (j) - 2 hat (k)

The position vectors of four points A,B,C and D are given below . In each case , using vector method prove that the four points A,B,C and D are coplanar . 6hat (i) - 4hat(j)+ 10 hat(k), - 5hat(i)+ 3hat(j)- 10 hat(k), 4 hat (i) - 6 hat (j) - 10 hat(k) , 2 hat (j) + 10 hat (k)