MAST10007 Linear Algebra
Practice class 5
Q1. Match the Cartesian equations in column 1, with the vector equations in column 2, and the
description in column 3.
Cartesian equation Vector equation Description
1. x
2= 3 −y=z+ 1
3A. r= (0,3,−1)
+s(3,0,−2) + t(0,3,1), s, t ∈Ri.
a line through the point
(2,17,3) and parallel
to v= (1,0,2)
2. 2x−y+ 3z=−6B. r = (2,17,3) + t(1,0,2), t ∈Rii.
a line through the point
(0,3,−1) and parallel
to v= (2,−1,3)
3. x−2 = z−3
2, y = 17 C. r = (0,3,−1) + t(2,−1,3), t ∈Riii.
a plane through the point
(0,3,−1) with normal
vector n= (2,−1,3)
Q2. Consider the straight line with Cartesian equation
x+ 1
3=y+ 2 = z−1
4
(a) Find a vector in the direction of the line.
(b) Does the point P(2,−1,5) lie on the line?
(c) Determine a vector equation for the line.
Q3. Determine the equation of the line which passes through (0,0,−1) and (1,0,−2) in vector,
parametric and Cartesian form.
Q4. Consider the plane with Cartesian equation
2x−3y+ 4z= 12
(a) Determine a vector perpendicular to the plane.
(b) Does the point (5,1,0) lie on the plane?
(c) Find a vector equation of the plane.
Q5. Determine the equation of the plane containing the points A(1,−1,1), B(2,0,2), and C(1,2,7)
in both vector and Cartesian form.
Q6. Consider the set V=R2with vector addition and scalar multiplication defined as
(x1, y1)+(x2, y2) = (x1+x2, y1+y2), α(x1, y1) = (2αx1,2αy1)
Decide which of the following two vector space axioms are satisfied. If the axiom holds, give
a proof. If not, give a counter-example. What can you conclude about whether Vis a vector
space?
Axioms:
For all u,v∈Vand α∈R,
M3: 1v=v
D1: α(u+v) = αu+αv
Mathematics and Statistics 1 University of Melbourne
