AA HL Vectors [83 marks]
1a. [4 marks]
The plane 𝛱1 has equation 3𝑥 − 𝑦 + 𝑧 = −13 and the line 𝐿 has vector equation
1
−3
𝐫 = (2 ) + 𝜆 (−1) , 𝜆 ∈ ℝ.
−2
4
Given that 𝐿 meets 𝛱1 at the point P, find the coordinates of P.
1b. [4 marks]
Find the shortest distance from the point O(0, 0, 0) to 𝛱1 .
1c. [3 marks]
The plane 𝛱2 contains the point O and the line 𝐿.
Find the equation of 𝛱2 , giving your answer in the form 𝐫. 𝐧 = 𝑑.
1d. [5 marks]
Determine the acute angle between 𝛱1 and 𝛱2 .
2a. [3 marks]
Points A(0 , 0 , 10) , B(0 , 10 , 0) , C(10 , 0 , 0) , V(𝑝 , 𝑝 , 𝑝) form the vertices of a tetrahedron.
10 − 2𝑝
→
→
Show that AB × AV = −10 (𝑝
) and find a similar expression for AC × AV.
𝑝
→
→
2b. [5 marks]
𝑝(3𝑝−20)
Hence, show that, if the angle between the faces ABV and ACV is 𝜃, then cos 𝜃 = 6𝑝2 −40𝑝+100.
2c. [3 marks]
Consider the case where the faces ABV and ACV are perpendicular.
Find the two possible coordinates of V.
2d. [1 mark]
Comment on the positions of V in relation to the plane ABC.
2e. [3 marks]
The following diagram shows the graph of 𝜃 against 𝑝. The maximum point is shown by X.
At X, find the value of 𝑝 and the value of 𝜃.
2f. [2 marks]
Find the equation of the horizontal asymptote of the graph.
3. [5 marks]
Three planes have equations:
2𝑥 − 𝑦 + 𝑧 = 5
𝑥 + 3𝑦 − 𝑧 = 4
, where 𝑎, 𝑏 ∈ ℝ.
3𝑥 − 5𝑦 + 𝑎𝑧 = 𝑏
Find the set of values of 𝑎 and 𝑏 such that the three planes have no points of intersection.
4. [6 marks]
5
5
A straight line, 𝐿𝜃 , has vector equation r = (0) + 𝜆 (sin 𝜃 ) , 𝜆, 𝜃 ∈ ℝ.
0
cos 𝜃
The plane 𝛱𝑝 , has equation 𝑥 = 𝑝, 𝑝 ∈ ℝ.
Show that the angle between 𝐿𝜃 and 𝛱𝑝 is independent of both 𝜃 and 𝑝.
5a. [5 marks]
Consider a triangle OAB such that O has coordinates (0, 0, 0), A has coordinates (0, 1, 2) and
B has coordinates (2𝑏, 0, 𝑏 − 1) where 𝑏 < 0.
Find, in terms of 𝑏, a Cartesian equation of the plane Π containing this triangle.
5b. [3 marks]
Let M be the midpoint of the line segment [OB].
Find, in terms of 𝑏, the equation of the line L which passes through M and is perpendicular
to the plane П.
5c. [7 marks]
Show that L does not intersect the 𝑦-axis for any negative value of 𝑏.
6a. [1 mark]
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.
Explain why ABCD is a parallelogram.
6b. [3 marks]
→
→
Using vector algebra, show that AD = BC.
6c. [5 marks]
→
→
→
→
The position vectors OA, OB, OC 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.
Show that p = 1, q = 1 and r = 4.
6d. [4 marks]
Find the area of the parallelogram ABCD.
6e. [4 marks]
The point where the diagonals of ABCD intersect is denoted by M.
Find the vector equation of the straight line passing through M and normal to the plane
𝛱 containing ABCD.
6f. [3 marks]
Find the Cartesian equation of 𝛱.
6g. [2 marks]
The plane 𝛱 cuts the x, y and z axes at X , Y and Z respectively.
Find the coordinates of X, Y and Z.
6h. [2 marks]
Find YZ.