Download Vectors 1 - Core 4 Revision 1. The lines L and M have vector

Vectors 1 - Core 4 Revision
1.
The lines L and M have vector equations
 1   p
   
L : r =  3  + t  3  and
 − 2  5 
   
 2
 2
 
 
M : r = q + s  1 
 3
 − 1
 
 
where p and q are constants.
(a)
Given that L and M are perpendicular, find the value of p.
(4)
(b)
Given also that L and M intersect, find the value of q.
(4)
(Total 8 marks)
2.
(a)
Find the exact value of the cosine of the acute angle between the two lines
 x   5
 1
   
 
 y  =  4  + λ  – 1
 z   – 2
 1
   
 
and
x  1
 5
   
 
 y  =  2 + µ  1 
 z   2
 1
   
 
(4)
(b)
Prove that the two lines do not intersect.
(4)
(Total 8 marks)
3.
The points A and B have coordinates (2, 4, 1) and (3, 2, –1) respectively. The point C is such
that OC = 2 OB , where O is the origin.
(a)
Find the vectors:
(i)
OC ;
(1)
(ii)
AB .
(2)
South Wolds Comprehensive School
1
(b)
(i)
Show that the distance between the points A and C is 5.
(2)
(ii)
Find the size of angle BAC, giving your answer to the nearest degree.
(4)
(c)
The point P(α, β, γ) is such that BP is perpendicular to AC.
Show that 4α – 3γ = 15.
(3)
(Total 12 marks)
4.
The points A and B have coordinates (1, 4, 2) and (2, –1, 3) respectively.
 2
 1


The line l has equation r =  − 1  + λ  − 1
 3
 1
(a)
Show that the distance between the points A and B is 3 3 .
(2)
(b)
The line AB makes an acute angle θ with l. Show that cos θ = 7 .
9
(3)
(c)
The point P on the line l is where λ = p.
(i)
Show that
 1
AP.  − 1 = 7 + 3 p
 1
(4)
(ii)
Hence find the coordinates of the foot of the perpendicular from the point A to the
line l.
(3)
(Total 12 marks)
5.
Find the acute angle between the lines
1 
1 
 
 
r =  2 + t  0
3
1 
 
 
and
1 
1 
 
 
r =  2 + s  2 .
 2
3
 
 
(Total 5 marks)
South Wolds Comprehensive School
2
6.
The lines l1 and l2 have vector equations r = (2 + λ)i + (–2 – λ)j + (7 + λ)k and
r = (4 + 4µ)i + (26 + 14µ)j + (–3 – 5µ)k respectively, where λ and µ are scalar parameters.
(a)
The vector n = –i + aj + bk where a and b are integers, is perpendicular to both lines.
Find the value of a and the value of b.
(3)
(b)
The point P on l1 and the point Q on l2 are such that PQ = m n for some scalar
constant m.
(i)
Determine the value of m.
(5)
(ii)
Deduce the shortest distance between l1 and l2.
(1)
(Total 9 marks)
7.
The points A, B and C have position vectors a = 4i – j – 2k, b = 3j - 8k and c = 11i + 2j + k
respectively.
(a)
Evaluate the scalar product (b – a) . (c – a).
(2)
(b)
Use this result to find, to the nearest 0.1° the size of angle BAC.
(3)
(Total 5 marks)
8.
The points A and B have position vectors a and b, respectively, relative to an origin O. The
triangle OAB is right-angled at A with a.b = 6 and |b| = 7.
B
b
θ
O
Given that angle AOB is θ, show that cos θ =
a
A
6
7
(Total 6 marks)
South Wolds Comprehensive School
3
9.
 2
1
 
 
The line l1 has equation r =  − 1  + s  2  .
 − 2
 4
 
 
7
 
The line l\ has equation r =  5  + t
 4
 
(a)
(i)
 2
 
 2 .
1
 
Find the value of the scalar product
 1   2
   
 2 •  2
 4  1 
   
(1)
(ii)
Show that the acute angle between the lines l1 and l2 is 43°, correct to the nearest
degree.
(3)
(b)
The line l1 intersects the plane x + y + z = 20 at the point Q. Find the position vector of
Q.
(3)
(Total 7 marks)
South Wolds Comprehensive School
4