AHL 3.

12 Vector Definitions
[248 marks]

The points A(5, −2, 5), B(5, 4, −1), C(−1, −2, −1) and D(7, −4, −3) are
the vertices of a right-pyramid.

1a. −−→ −−→ [2 marks]

Find the vectors AB and AC .

1b. Use a vector method to show that BÂC = 60°. [3 marks]

1c. Show that the Cartesian equation of the plane Π that contains the [3 marks]
triangle ABC is −x + y + z = −2.

The line L passes through the point D and is perpendicular to Π .

1d. Find a vector equation of the line L. [1 mark]

D
1e. Hence determine the minimum distance, dmin , from D to Π . [4 marks]

ABCD
1f. Find the volume of right-pyramid ABCD. [4 marks]
 12
Consider the vectors a and b such that a =( ) and |b|= 15.
−5

2a. Find the possible range of values for |a + b| . [2 marks]

Consider the vector p such that p = a + b.

2b. Given that |a + b| is a minimum, find p. [2 marks]

 =( ), where x, y ∈ R+ .
x
Consider the vector q such that q
y

2c. Find q such that |q| = |b| and q is perpendicular to a. [5 marks]

O
 Two airplanes, A and B, have position vectors with respect to an origin O given
respectively by

⎛ 19 ⎞ ⎛ −6 ⎞
rA =⎜ −1 ⎟+t⎜ 2 ⎟
⎝ 1 ⎠ ⎝ 4 ⎠

⎛1 ⎞ ⎛ 4 ⎞
rB =⎜ 0 ⎟+t⎜ 2 ⎟
⎝ 12 ⎠ ⎝ −2 ⎠
where t represents the time in minutes and 0 ≤ t ≤ 2. 5.
Entries in each column vector give the displacement east of O , the displacement
north of O and the distance above sea level, all measured in kilometres.

3a. Find the three-figure bearing on which airplane B is travelling. [2 marks]

3b. Show that airplane A travels at a greater speed than airplane B. [2 marks]

3c. Find the acute angle between the two airplanes’ lines of flight. Give your [4 marks]
answer in degrees.

P
 The two airplanes’ lines of flight cross at point P .

3d. Find the coordinates of P. [5 marks]

3e. Determine the length of time between the first airplane arriving at P and [2 marks]
the second airplane arriving at P .
3f. LetD(t) represent the distance between airplane A and airplane B for [5 marks]
0 ≤ t ≤ 2. 5.
Find the minimum value of D(t).

A(3, 0, 0), B(0, − 2, 0) C(1, 1, − 7)

 Three points A(3, 0, 0), B(0, − 2, 0) and C(1, 1, − 7) lie on the plane Π1 .

4a. −−→ −−→ [2 marks]

Find the vector AB and the vector AC .
4b. Hence find the equation of Π 1 , expressing your answer in the form [5 marks]
ax + by + cz = d, where a, b, c, d ∈ Z.

3 − +2 =2
 Plane Π2 has equation 3x − y + 2z = 2.

4c. The line

L is the intersection of Π1 and Π2 . Verify that the vector [2 marks]
⎛ 0 ⎞ ⎛ 1 ⎞
equation of L can be written as r = ⎜ −2 ⎟+λ⎜ 1 ⎟.
⎝ 0 ⎠ ⎝ −1 ⎠

The plane Π 3 is given by 2x − 2z = 3. The line L and the plane Π3 intersect at

the point P.

4d. Show that at the point P, λ = 34 . [2 marks]

P
4e. Hence find the coordinates of P. [1 mark]

The point B(0, −2, 0) lies on L.

4f. Find the reflection of the point B in the plane Π 3 . [7 marks]

4g. Hence find the vector equation of the line formed when L is reflected in [2 marks]
the plane Π 3 .

Points A and B have coordinates (1, 1, 2) and (9, m, − 6) respectively.

5a. −−→ [2 marks]

Express AB in terms of m.

⎛ −3 ⎞ ⎛ 2 ⎞
 ⎛ −3 ⎞ ⎛ 2 ⎞
The line L, which passes through B, has equation r = ⎜ −19 ⎟+s⎜ 4 ⎟.
⎝ 24 ⎠ ⎝ −5 ⎠

5b. Find the value of m . [5 marks]

2 1
5c. Consider a unit vector u, such that u = pi − 23 j + 13 k, where p > 0. [8 marks]
−−→
Point C is such that BC = 9u.
Find the coordinates of C.

⎛ −2 ⎞ ⎛6⎞
 ⎛ −2 ⎞ ⎛6⎞
The points A and B have position vectors ⎜ 4 ⎟ and ⎜ 8 ⎟ respectively.
⎝ −4 ⎠ ⎝0⎠

⎛ −1 ⎞
Point C has position vector ⎜ k ⎟. Let O be the origin.
⎝ 0 ⎠

Find, in terms of k,

6a. −−→ −−→ [2 marks]

OA ∙ OC .

6b. −
−→ −−→ [1 mark]
OB ∙ OC .

ˆ ˆ
6c. Given that AÔC = BÔC, show that k = 7. [8 marks]

AOC
6d. Calculate the area of triangle AOC. [6 marks]
 Consider the lines L1 and L2 with respective equations
L1 : y = − 23 x + 9 and L2 : y = 25 x − 19
5
.

7a. Find the point of intersection of L 1 and L2 . [2 marks]

A third line, L3 , has gradient − 34 .

7b. Write down a direction vector for L 3 . [1 mark]

7c. L3 passes through the intersection of L1 and L2 . [2 marks]

Write down a vector equation for L 3 .

⎛0⎞ ⎛0 ⎞
 ⎛0⎞ ⎛0 ⎞
Consider the vectors a = ⎜ 3 ⎟ and b = ⎜ 6 ⎟.
⎝p⎠ ⎝ 18 ⎠
Find the value of p for which a and b are

8a. parallel. [2 marks]

8b. perpendicular. [4 marks]

P P
 Two distinct lines, l1 and l2 , intersect at a point P . In addition to P , four distinct
points are marked out on l1 and three distinct points on l2 . A mathematician
decides to join some of these eight points to form polygons.

9a. Find how many sets of four points can be selected which can form the [2 marks]
vertices of a quadrilateral.
9b. Find how many sets of three points can be selected which can form the [4 marks]
vertices of a triangle.

⎛1⎞ ⎛1⎞
 ⎛1⎞ ⎛1⎞
The line l1 has vector equation r1 = ⎜ 0 ⎟ + λ ⎜ 2 ⎟, λ ∈ R and the line l2 has
⎝1⎠ ⎝1⎠

⎛ −1 ⎞ ⎛5⎞
vector equation r2 = ⎜ 0 ⎟ + μ ⎜ 6 ⎟, μ ∈ R.
⎝ 2 ⎠ ⎝2⎠
The point P has coordinates (4, 6, 4).

9c. Verify that P is the point of intersection of the two lines. [3 marks]

The point A has coordinates (3, 4, 3) and lies on l1 .

9d. Write down the value of λ corresponding to the point A. [1 mark]

B
 The point B has coordinates (−1, 0, 2) and lies on l2 .

9e. −−→ −→ [2 marks]

Write down PA and PB .

C D
9f. Let C be the point on l1 with coordinates (1, 0, 1) and D be the point on [8 marks]
l2 with parameter μ = −2.
Find the area of the quadrilateral CDBA.
 Three points in three-dimensional space have coordinates A(0, 0, 2), B(0, 2, 0) and
C(3, 1, 0).

10a. −−→ [1 mark]

Find the vector AB .

10b. −−→ [1 mark]

Find the vector AC .
10c. Hence or otherwise, find the area of the triangle ABC. [4 marks]

Consider the points A(−3, 4, 2) and B(8, −1, 5).

11a. −−→ [2 marks]

Find AB .
11b. ∣−−→∣ [2 marks]
Find ∣AB ∣.
∣ ∣

⎛ 2 ⎞ ⎛ 1 ⎞
A line L has vector equation r = ⎜ 0 ⎟ + t ⎜ −2 ⎟. The point C (5, y, 1) lies on
⎝ −5 ⎠ ⎝ 2 ⎠
line L.

11c. Find the value of y. [3 marks]

⎛ 8 ⎞
 −−→ ⎛
11d. 8 ⎞ [2 marks]
Show that AC = ⎜ −10 ⎟.
⎝ −1 ⎠

11e. −−→ −−→ [5 marks]

Find the angle between AB and AC .
11f. Find the area of triangle ABC. [2 marks]

Point A has coordinates (−4, −12, 1) and point B has coordinates (2, −4, −4).

12a.
⎛ 6 ⎞ [1 mark]
→ =
Show that AB
⎝ −5 ⎠
8
 The line L passes through A and B.

12b. Find a vector equation for L. [2 marks]

12c. Point C (k , 12 , −k) is on L. Show that k = 14. [4 marks]

12d. Find → ∙ AB
OB → . [2 marks]
12e. Write down the value of angle OBA. [1 mark]

12f. Point D is also on L and has coordinates (8, 4, −9). [6 marks]

Find the area of triangle OCD.
 The points A, B, C and D have position vectors a, b, c and d, relative to the origin
O.
→
It is given that AB → .
= DC

13a. Explain why ABCD is a parallelogram. [1 mark]

13b. Using vector algebra, show that AD

→ → .
= BC [3 marks]
 → ,
The position vectors OA → , OC
OB → and OD
→ are given by
a = i + 2j − 3k
b = 3i − j + pk
c = qi + j + 2k
d = −i + rj − 2k
where p , q and r are constants.

13c. Show that p = 1, q = 1 and r = 4. [5 marks]

13d. Find the area of the parallelogram ABCD. [4 marks]

The point where the diagonals of ABCD intersect is denoted by M.

13e. Find the vector equation of the straight line passing through M and [4 marks]
normal to the plane Π containing ABCD.

Π
13f. Find the Cartesian equation of Π. [3 marks]

The plane Π cuts the x, y and z axes at X , Y and Z respectively.

13g. Find the coordinates of X, Y and Z. [2 marks]

13h. Find YZ. [2 marks]

Two submarines A and B have their routes planned so that their positions at time t
hours, 0 ≤ t < 20 , would be defined by the position vectors rA
⎛ 2 ⎞ ⎛ −1 ⎞ ⎛ 0 ⎞ ⎛ −0.5 ⎞
=
⎝ −1 ⎠
+t
⎝ −0.15 ⎠
and rB =
⎝ −2 ⎠
3.2 + t
⎝ 0.1 ⎠
4 1 1.2 relative to a fixed

point on the surface of the ocean (all lengths are in kilometres).

14a. Show that the two submarines would collide at a point P and write down [4 marks]
the coordinates of P.
 To avoid the collision submarine B adjusts its velocity so that its position vector is
now given by

⎛ 0 ⎞ ⎛ −0.45 ⎞
rB =
⎝ −2 ⎠
3.2 + t
⎝ 0.09 ⎠
1.08 .

14b. Show that submarine B travels in the same direction as originally [1 mark]
planned.

14c. Find the value of t when submarine B passes through P. [2 marks]

14d. Find an expression for the distance between the two submarines in [5 marks]
terms of t.

14e. Find the value of t when the two submarines are closest together. [2 marks]
14f. Find the distance between the two submarines at this time. [1 mark]

−−→ −−→
In the following diagram, OA = a, OB = b. C is the midpoint of [OA] and
−−→ 1 −→
OF = 6 FB .

15a. −−→ [1 mark]

Find, in terms of a and b OF .

−−→
15b. −−→ [2 marks]
Find, in terms of a and b AF .

−−→ −−→ −−→ −−→

It is given also that AD = λAF and CD = μCB , where λ, μ ∈ R.

15c. −−→ [2 marks]

Find an expression for OD in terms of a, b and λ;

−−→
15d. −−→ [2 marks]
Find an expression for OD in terms of a, b and μ.

15e. Show that μ 1 [4 marks]

= 13
, and find the value of λ.

−−→
15f. −−→ [2 marks]
Deduce an expression for CD in terms of a and b only.

ΔOAB = (area ΔCAD)

15g. Given that area ΔOAB = k(area ΔCAD), find the value of k. [5 marks]

⎛4⎞
 −−→ ⎛ ⎞
4
Let AB = ⎜ 1 ⎟.
⎝2⎠

16a. ∣−−→∣ [2 marks]

Find ∣AB ∣.
∣ ∣

−−→ ⎛ ⎞
16b. 3 [4 marks]
Let AC = ⎜ 0 ⎟. Find BAC
^ .
⎝0⎠

A line L1 passes through the points A(0, 1, 8) and B(3, 5, 2).

17a. −−→ [2 marks]

Find AB .
17b. Hence, write down a vector equation for L1 . [2 marks]

⎛ 1 ⎞ ⎛p⎞
17c. [3 marks]
A second line L2 , has equation r = ⎜ 13 ⎟ + s ⎜ 0 ⎟.
⎝ −14 ⎠ ⎝1⎠
Given that L1 and L2 are perpendicular, show that p = 2.

=2
 Given that L1 and L2 are perpendicular, show that p = 2.

17d. The lines L1 and L1 intersect at C(9, 13, z). Find z. [5 marks]

17e. Find a unit vector in the direction of L 2 . [2 marks]

√5
 17f. Hence or otherwise, find one point on L 2 which is √5 units from C. [3 marks]

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