## Scalar Product and the Ratio Theorem Exercise

1. Calculate the angles between the following pairs of vectors, giving your answers in degrees to 1 decimal place, where not exact.

 a b c d e f 2. Find the angle between the line joining the points A = (2, 3) and B = (4, -6) and the line joining the points C = (3, -4) and D = (2, 5)

3. Find the angle between the line joining the points E = (2, 3, -4) and F = (3, 6, -2) and the line joining the points G = (-2, 5, 4) and H = (4, 3, 2)

4. Find the coordinates of the point P where P lies on the line AB such that:

 a A = (3, 6) and B = (-3, 3) AP:PB = 2:1 b A = (6, 1, 5) and B = (2, 5, -3) AP:PB = 1:3 c A = (4, -3, 1) and B = (-1, 2, -4) AP:PB = 3:2 d A = (-4, 5) and B = (3, -9) AP/PB = 3/4 e A = (-3, 8) and B = (4, 4) AP:PB = -2:1 f A = (6, -2, -1) and B = (2, 0 , 7) AP:PB = 3: -5

5. The diagram shows a triangular prism with a horizontal rectangular base ADFC, where CF = 40 units and AC = 14 units.
The vertical ends ABC and DEF are isosceles triangles with AB = BC = 25 units.
The midpoints of BE and DF are M and N respectively.
The origin O is at the mid-point of AC.
Unit vectors i, j and are parallel to OC, ON and OB respectively.

(i) Find the length of OB.

(ii) Express MC and MN in terms of i, j and k

(iii) Evaluate MC.MN and hence find angle CMN giving your answer to the nearest degree.

6. Given that a = and b = and c = find

a. The angle between the directions of and b.

b. The value of p for which and are perpendicular.

7. The diagram shows a prism with cross-section in the shape of a right-angled triangle OAC where BE = 16 cm and OC = 26 cm. The cross-section through E is the triangle BED. The length of the prism is 25 cm. M is the mid-point of CD and N is the mid-point of DE.
Unit vectors i, j and are parallel to OA, OB and OC respectively as shown.

(i) Express vectors MN and MA in terms of i, j and k.
(ii) Evaluate the angle NMA giving your answer to the nearest degree.