Vector Space and Equation
HL 3D Geo Test #1eCalculators PermittedName________________________ Time: 70 minutes100 points
1) Solve the system of simultaneous linear equations .
2)Find the equation of the line that is perpendicular to the line with equation and that passes through the point with coordinates (2, 1). What is the perpendicular distance from the origin to the line with equation ?
3) Solve the inequality 2
4)Consider the vectors a = i − j + k, b = i + 2 j + 4k and c = 2i − 5 j − k.
(a)Given that c = ma + nb where m, n , find the value of m and of n.
(5)
(b)Find a unit vector, u, normal to both a and b.
(5)
(c)The plane 1 contains the point A (1, –1, 1) and is normal to b. The plane intersects the x, y and z axes at the points L, M and N respectively.
(i)Find a Cartesian equation of 1.
(ii)Write down the coordinates of L, M and N.
(5)
(d)The line through the origin, O, normal to π1 meets π1 at the point P.
(i)Find the coordinates of P.
(ii)Hence find the distance of π1 from the origin.
(7)
(e)The plane 2 has equation x + 2y + 4z = 4. Calculate the angle between 2 and a line parallel to a.
(5)
(Total 27 marks) 5)Two planes 1 and 2 are represented by the equations
1: r =
2: 2x – y – 2z = 4.
(a)(i)Find
(ii)Show that the equation of 1 can be written as x − 2y + 2z =11.
(4)
(b)Show that 1 is perpendicular to 2.
(4)
(c)The line l1 is the line of intersection of 1 and 2.
Find the vector equation of l1, giving the answer in parametric form.
(5)
(d)The line l2 is parallel to both 1 and 2, and passes through P(3, –5, –1).
Find an equation for l2 in Cartesian form.
(3)
(e)Let Q be the foot of the perpendicular from P to the plane 2.
(i)Find the coordinates of Q.
(ii)Find PQ.
(7)
(Total 23 marks)
BONUS
(a)The plane π1 has equation r = .
The plane π2 has the equation r = .
(i)For points which lie in π1 and π2, show that, = .
(ii)Hence, or otherwise, find a vector equation of the line of intersection of π and π2.
1) Solve the system of simultaneous linear equations .
2)Find the equation of the line that is perpendicular to the line with equation and that passes through the point with coordinates (2, 1). What is the perpendicular distance from the origin to the line with equation ?
3) Solve the inequality 2
4)Consider the vectors a = i − j + k, b = i + 2 j + 4k and c = 2i − 5 j − k.
(a)Given that c = ma + nb where m, n , find the value of m and of n.
(5)
(b)Find a unit vector, u, normal to both a and b.
(5)
(c)The plane 1 contains the point A (1, –1, 1) and is normal to b. The plane intersects the x, y and z axes at the points L, M and N respectively.
(i)Find a Cartesian equation of 1.
(ii)Write down the coordinates of L, M and N.
(5)
(d)The line through the origin, O, normal to π1 meets π1 at the point P.
(i)Find the coordinates of P.
(ii)Hence find the distance of π1 from the origin.
(7)
(e)The plane 2 has equation x + 2y + 4z = 4. Calculate the angle between 2 and a line parallel to a.
(5)
(Total 27 marks) 5)Two planes 1 and 2 are represented by the equations
1: r =
2: 2x – y – 2z = 4.
(a)(i)Find
(ii)Show that the equation of 1 can be written as x − 2y + 2z =11.
(4)
(b)Show that 1 is perpendicular to 2.
(4)
(c)The line l1 is the line of intersection of 1 and 2.
Find the vector equation of l1, giving the answer in parametric form.
(5)
(d)The line l2 is parallel to both 1 and 2, and passes through P(3, –5, –1).
Find an equation for l2 in Cartesian form.
(3)
(e)Let Q be the foot of the perpendicular from P to the plane 2.
(i)Find the coordinates of Q.
(ii)Find PQ.
(7)
(Total 23 marks)
BONUS
(a)The plane π1 has equation r = .
The plane π2 has the equation r = .
(i)For points which lie in π1 and π2, show that, = .
(ii)Hence, or otherwise, find a vector equation of the line of intersection of π and π2.