Time allowed: 15 minutes.
1. (2 marks)
Given
A
=
⎛
⎝
7
0
−
1
⎞
⎠
,B
=
⎛
⎝
3
3
3
⎞
⎠
,C
=
⎛
⎝
4
−
4
2
⎞
⎠
, ﬁnd the coordinates of the point
D
=
⎛
⎝
x
y
z
⎞
⎠
such that
ABCD
is
a parallelogram.
2. (2 marks)
Let
A
=
⎛
⎝
6
−
1
−
3
⎞
⎠
=
⎛
⎝
15
2
−
6
⎞
⎠
. Find in parametric vector form the equation of the line through
A
and
B
.
Is this line parallel to the line
˜
x
=
⎛
⎝
0
12
−
27
⎞
⎠
+
λ
⎛
⎝
3
1
−
1
⎞
⎠
? Give a reason for your answer.
3. (3 marks)
Let
S
=
⎧
⎨
⎩
˜
x
∈
R
3
:
˜
x
=
λ
⎛
⎝
2
1
3
⎞
⎠
+
µ
⎛
⎝
2
1
−
1
⎞
⎠
for 0
≤
λ
≤
2 and
λ
≤
µ
≤
3
λ
⎫
⎬
⎭
Neatly sketch
S
, labelling all important points.
4. (3 marks)
For the following system of equations, write down the corresponding augmented matrix, use Gaussian elimi
nation to transform the augmented matrix into rowechelon form, and solve the system of equations, writing
your answer in vector form.
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 Three '11
 15 minutes, DC., 95 Week, 0 1 Week

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