6. Titik-titik \( V(k, 2 k), W(4 a, 2 b) \) dan \( X(8 a, 6 b) \) terletak pada suatu garis lurus. Titik \( W \) membahagi garis \( V X \) dengan nisbah \( 1: 4 \). Tunjukkan bahawa \( a=\frac{1}{6} b \). (TP4 Points \( V(k, 2 k), W(4 a, 2 b) \) and \( X(8 a, 6 b) \) lie on a straight line. Point \( W \) divides line \( V X \) with the ratio of \( 1: 4 \). Show that \( a=\frac{1}{6} b \).

1. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then use the algebra of vectors find the followings \[ \begin{array}{l} (3 u-4 v) \bullet(5 w-2 u) \\ (2 v+3 w) \bullet(4 u-3 v) \\ (u+v+3 w) \bullet(u-v-2 w) \\ (2 u+v) \times(v-2 w) \\ (w+u) \times(w-u) \\ (v+u) \times(2 w-3 u) \end{array} \] 2. Find a vector perpendicular to the plane of using the points \( P=(2,4,6) \), \( Q=(1,2,9) \) and \( R=(1,-1,7) \). Also find area of the parallelogram and area of triangle. Furthermore find a unit vector perpendicular to the plane using these points. 3. Find a vector perpendicular to the plane of using the points \( P=(1,2,-4) \), \( Q=(2,5,-7) \) and \( R=(3,-5,-1) \). Also find area of the parallelogram and area of triangle. Furthermore find a unit vector perpendicular to the plane using these points. 4. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then show that \( (u \times v) \bullet w=(v \times w) \bullet u=(w \times u) \bullet v \) 5. If \( u=2 i+4 j+8 k, v=2 i-3 j-6 k \) and \( w=5 i+4 j+8 k \) then show that \( (u \times v) \bullet w=(v \times w) \bullet u=(w \times u) \bullet v \) 6. Find the volume of the box (parallelepiped) determined by \( u=4 i+5 j-2 k \), \( v=2 i+3 j+5 k \) and \( w=5 i+4 j+8 k \) 7. Find the volume of the box (parallelepiped) determined by \( u=2 i+6 j-3 k \), \( v=5 i-4 j-7 k \) and \( w=-7 i-4 j+2 k \)

3. Find a vector perpendicular to the plane of using the points \( P=(1,2,-4) \), \( Q=(2,5,-7) \) and \( R=(3,-5,-1) \). Also find area of the parallelogram and area of triangle. Furthermore find a unit vector perpendicular to the plane using these points.

Figure 4.27 The areas of two similar triangles are \( 64 \mathrm{~cm}^{2} \) and \( 144 \mathrm{~cm}^{2} \), a) Find the ratio of their altitudes. b) If the perimeter of the smaller triangle is 49 cm , find the perimeter of the larger triangle.

6. Find the volume of the box (parallelepiped) determined by \( u=4 i+5 j-2 k \), \( v=2 i+3 j+5 k \) and \( w=5 i+4 j+8 k \) 7. Find the volume of the box (parallelepiped) determined by \( u=2 i+6 j-3 k \), \( v=5 i-4 j-7 k \) and \( w=-7 i-4 j+2 k \)

8. Find the volume of the box (parallelepiped) determined by the points \( P=(1,2,4) \), \( Q=(2,-5,8), R=(4,2,-3) \) and \( S=(3,5,-6) \). 9. Find the volume of the box (parallelepiped) determined by the points \( P=(8,5,3) \), \( Q=(4,6,7), R=(-4,-2,9) \) and \( S=(3,-3,-4) \).