The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1002: Linear Algebra Semester 2, 2023
Lecturer: Brad Roberts
This individual assignment is due by 11:59pm Tuesday 22 August 2023, via
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Copyright © 2023 The University of Sydney 1
1. Let s, t ∈ R, and consider the vectors
u =
20
−1
, v =
1−2
−2
, w =
118
s
and x =
t + 32t− 4
3t− 8
.
(a) Compute the area of the triangle inscribed by the vectors u and v.
(b) Find all values of s such that w is a linear combination of u and v.
(c) Find all values of t for which v and x are orthogonal.
(d) Compute the projection of the vector x onto v; your answer should be a vector
whose components are expressions in terms of t.
(e) Find a point P in R3 such that the points (0, 0, 0), (1,−2,−2), (6, 2, 1), and P
form a rectangle. Explain your reasoning.
2. Suppose that a,b ∈ R3 are two vectors with the following properties:
b =
34
0
.
a · b = 10.
The angle between a and b is 60◦.
(a) Find the length of a.
(b) Suppose in addition that a =
xy
z
, with y = 1. What are all possible values of x
and z?
3. Consider the line ` in R2 with normal vector n =
[
1
−5
]
and passing through the point
P = (−3, 4).
(a) Write down equations in normal form and general form for this line.
(b) Use the general form to find parametric equations for `, and then write down a
vector equation for ` as well.
(c) Hence or otherwise write down a direction vector for `.