1. Calculate the angles between the following pairs of vectors, giving your answers in degrees to 1 decimal place, where not exact.
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a
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b
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c
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d
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e
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f
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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:
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a
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A = (3, 6) and B = (-3, 3) |
AP:PB = 2:1
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b
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A = (6, 1, 5) and B = (2, 5, -3) |
AP:PB = 1:3
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c
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A = (4, -3, 1) and B = (-1, 2, -4) |
AP:PB = 3:2
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d
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A = (-4, 5) and B = (3, -9) |
AP/PB = 3/4
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e
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A = (-3, 8) and B = (4, 4) |
AP:PB = -2:1
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f
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A = (6, -2, -1) and B = (2, 0 , 7) |
AP:PB = 3: -5
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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 k 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 a and b.
b. The value of p for which b and c 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 k 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.
