Show that the following vectors are linearly dependent
`bar(i)+bar(j), bar(j)+bar(k), -bar(k)+bar(i)`

Show that the following vectors are linearly dependent i) bar(i)+bar(j)+2bar(k), bar(j)+2bar(k), bar(i)+2bar(j)+4bar(k)

Compute [bar(i) - bar(j). bar(j) - bar(k). bar(k) -bar(i)] .

Show that the four points with position vectors 3bar(i)+5bar(j), 3bar(j)-5k, 5bar(i)-19bar(j)-3bar(k), 6bar(i)-5bar(j) are non coplanar.

Show that the triangle formed by the vectors 3bar(i)+5bar(j)+2bar(k), 2bar(i)-3bar(j)-5bar(k), -5bar(i)-2bar(j)+3bar(k) is equilateral.

If bar(i), bar(j), bar(k) are unit vectors along the positive directions of the coordinate axes, then shown that the four points 4 bar(i) + 5 bar(j) + bar(k), - bar(j) - bar(k), 3 bar(i) +9 bar(j) + 4 bar(k) and - 4 bar(i) + 4 bar(j) + 4 bar( k) are coplanar

Find unit vector perpendicular to both bar(i) + bar(j) + bar(k) and 2bar(i) + bar(j) + 3bar(k) .

Find a unit vector perpendicular to the plane containing the vector bar(a) = 4bar(i) + 3bar(j) - bar(k), bar(b) = 2bar(i) - 6bar(j) - 3bar(k)

Write the position vector of the centroid of the triangle formed by the points whose position vectors are 3bar(i)+2bar(j)-bar(k), 2bar(i)-2bar(j)+5bar(k), bar(i)+3bar(j)-bar(k) .

Find the volume of the tetrahedron having the edges , bar(i) + bar(j) + bar(k), bar(i)-bar(j), bar(i) + 2bar(j) + bar(k) .

The vectors 2bar(i)-3bar(j)+bar(k), bar(i)-2bar(j)+3bar(k), 3bar(i)+bar(j)-2bar(k)