St Leonard’s College
IBHL Mathematics
Vectors Test
May 2017
Section A: Calculator Free
Question 1:[Maximum mark:4]
Consider the vectors and
a)Find:
i)
ii)
[2marks]
Let , where 0 is the zero vector
b)Find
[2marks]
Question 2: [Maximum mark:6]
Consider the points A (1, 2 , 3) , B(1, 0, 5) and C(2 , -1, 4) .
a)Find
[4marks]
b)Hence find the area of the triangle ABC
[2marks]
Question 3: [Maximum mark:3]
Find the coordinates of the point where the line given by the parametric equations
, , and intersects the plane with equation
2x + 3y – z = 2.
[3marks]
Question 4: [Maximum mark:4]
The vector = 2 – +3 is normal to a plane which passes through the point (2, 1, 2).
a) Find an equation for the plane.
b) Find a if the point (a, a – 1, a – 2) lies on the plane.
[4marks]
Question 5: [Maximum mark:21]
The vertices of triangle ABC have coordinates given by , and
a)i) Find the lengths of the sides of the triangle
iii)Find
[6marks]
b)i) Show that
iv)Hence show that the area of the triangle ABC is
[5marks]
c)Find the Cartesian equation of the plane containing triangle ABC
[3marks]
d)Find the vector equation of AB
[2marks]
The point D on AB is such that is perpendicular to where O is the origin.
e)i) Find the coordinates of D
ii)Show that D does not lie between AB
[5marks]
Question 6: [Maximum mark: 20]
Consider the points P(-3, -1, 2) and Q(5, 5, 6) .
a)Find a vector equation for the line , which passes through the points P and Q
[3marks]
The line , has equation
b)Show that and intersect at the point
[4marks]
c) Find the acute angle between and
[3marks]
Let S be the point on such that
d) Show that one of the possible positions for S is and find the coordinates of the other possible position
[6marks]
f)Find the vector equation of the line that passes through R and bisects PR
[4marks]