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Level 1
Level 2
Level 3
Level 4
Level 5
Column vectors
16 questions
Lesson
Not started
Mark as done
a
=
(
6
−
8
)
\mathbf{a} = \binom{6}{-8}
a
=
(
−
8
6
)
and
b
=
(
−
9
12
)
\mathbf{b} = \binom{-9}{12}
b
=
(
12
−
9
)
.
(a)
Work out
∣
a
∣
|\mathbf{a}|
∣
a
∣
.
(b)
Show that
b
\mathbf{b}
b
is parallel to
a
\mathbf{a}
a
.
(c)
Hence write down the value of
∣
b
∣
|\mathbf{b}|
∣
b
∣
.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
a
=
(
4
−
1
)
\mathbf{a} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}
a
=
(
4
−
1
)
\qquad
b
=
(
−
1
3
)
\mathbf{b} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}
b
=
(
−
1
3
)
The vector
(
t
+
2
3
t
)
\begin{pmatrix} t + 2 \\ 3t \end{pmatrix}
(
t
+
2
3
t
)
is parallel to the vector
a
+
2
b
\mathbf{a} + 2\mathbf{b}
a
+
2
b
Find the value of
t
t
t
.
(3 marks)
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A
B
→
=
(
6
−
4
)
\overrightarrow{AB} = \binom{6}{-4}
A
B
=
(
−
4
6
)
and
B
B
B
is the point
(
4
,
−
1
)
(4, -1)
(
4
,
−
1
)
.
(a)
Find the coordinates of
A
A
A
.
(b)
M
M
M
is the midpoint of
A
B
AB
A
B
. Find the coordinates of
M
M
M
.
●●●●
●
Level 4
4 marks
Start
→
Mark as done
a
=
(
3
−
1
)
\mathbf{a} = \binom{3}{-1}
a
=
(
−
1
3
)
and
b
=
(
−
2
5
)
\mathbf{b} = \binom{-2}{5}
b
=
(
5
−
2
)
.
(a)
Work out
2
a
−
3
b
2\mathbf{a} - 3\mathbf{b}
2
a
−
3
b
as a column vector.
(b)
The vector
(
k
20
)
\binom{k}{20}
(
20
k
)
is parallel to
b
\mathbf{b}
b
. Find the value of
k
k
k
.
●●●●
●
Level 4
5 marks
Start
→
Vectors on a shape
13 questions
Lesson
Not started
Mark as done
O
A
B
OAB
O
A
B
is a triangle.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
B
→
=
b
\overrightarrow{OB} = \mathbf{b}
O
B
=
b
.
P
P
P
is the point on
A
B
AB
A
B
such that
A
P
:
P
B
=
2
:
1
AP : PB = 2 : 1
A
P
:
P
B
=
2
:
1
.
M
M
M
is the midpoint of
O
B
OB
O
B
.
(a)
Find
O
P
→
\overrightarrow{OP}
O
P
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form. [3]
(b)
Find
P
M
→
\overrightarrow{PM}
P
M
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form. [2]
●●●●
●
Level 4
5 marks
Start
→
Mark as done
O
A
B
OAB
O
A
B
is a triangle.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
B
→
=
b
\overrightarrow{OB} = \mathbf{b}
O
B
=
b
.
The point
P
P
P
lies on
A
B
AB
A
B
so that
A
P
:
P
B
=
1
:
4
AP:PB = 1:4
A
P
:
P
B
=
1
:
4
.
a
b
O
A
B
P
The diagram is a sketch and is not accurately drawn.
Find
O
P
→
\overrightarrow{OP}
O
P
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
, giving your answer in its simplest form.
●●●●
●
Level 4
4 marks
Start
→
Mark as done
O
A
B
C
OABC
O
A
B
C
is a trapezium.
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
C
→
=
b
\overrightarrow{OC} = \mathbf{b}
O
C
=
b
.
C
B
CB
C
B
is parallel to
O
A
OA
O
A
, and
C
B
=
3
×
O
A
CB = 3 \times OA
C
B
=
3
×
O
A
.
M
M
M
is the midpoint of
A
B
AB
A
B
.
a
b
O
A
B
C
M
The diagram is a sketch and is not accurately drawn.
(a)
Find
O
B
→
\overrightarrow{OB}
O
B
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
.
(b)
Find
O
M
→
\overrightarrow{OM}
O
M
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
, giving your answer in its simplest form.
●●●●
●
Level 4
5 marks
Start
→
Vector proofs
12 questions
Lesson
Not started
Mark as done
O
A
B
OAB
O
A
B
is a triangle.
O
A
→
=
2
a
\overrightarrow{OA} = 2\mathbf{a}
O
A
=
2
a
and
O
B
→
=
2
b
\overrightarrow{OB} = 2\mathbf{b}
O
B
=
2
b
.
M
M
M
is the midpoint of
O
A
OA
O
A
and
N
N
N
is the midpoint of
O
B
OB
O
B
.
O
A
B
M
N
2a
2b
Diagram not accurately drawn.
(a)
Find
M
N
→
\overrightarrow{MN}
M
N
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form.
(b)
Prove that
M
N
MN
M
N
is parallel to
A
B
AB
A
B
.
(c)
Hence write down the ratio
M
N
:
A
B
MN : AB
M
N
:
A
B
.
●●●●
●
Level 4
6 marks
Start
→
Mark as done
P
Q
R
S
PQRS
P
QR
S
is a quadrilateral.
P
Q
→
=
5
a
+
2
b
\overrightarrow{PQ} = 5\mathbf{a} + 2\mathbf{b}
P
Q
=
5
a
+
2
b
,
Q
R
→
=
3
a
−
4
b
\quad\overrightarrow{QR} = 3\mathbf{a} - 4\mathbf{b}
QR
=
3
a
−
4
b
,
R
S
→
=
−
5
a
−
2
b
\quad\overrightarrow{RS} = -5\mathbf{a} - 2\mathbf{b}
R
S
=
−
5
a
−
2
b
.
(a)
Find
P
R
→
\overrightarrow{PR}
P
R
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form.
(b)
Prove that
P
Q
R
S
PQRS
P
QR
S
is a parallelogram.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
O
O
O
,
A
A
A
and
B
B
B
are three points, with
O
A
→
=
a
\overrightarrow{OA} = \mathbf{a}
O
A
=
a
and
O
B
→
=
b
\overrightarrow{OB} = \mathbf{b}
O
B
=
b
.
The point
C
C
C
is such that
O
C
→
=
1
5
a
+
4
5
b
\overrightarrow{OC} = \tfrac{1}{5}\mathbf{a} + \tfrac{4}{5}\mathbf{b}
O
C
=
5
1
a
+
5
4
b
.
(a)
Show that
C
C
C
lies on the line
A
B
AB
A
B
.
(b)
Write down the ratio
A
C
:
C
B
AC : CB
A
C
:
C
B
.
●●●●
●
Level 4
4 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a quadrilateral.
A
B
→
=
4
a
+
2
b
\overrightarrow{AB} = 4\mathbf{a} + 2\mathbf{b}
A
B
=
4
a
+
2
b
,
B
C
→
=
a
−
3
b
\quad\overrightarrow{BC} = \mathbf{a} - 3\mathbf{b}
B
C
=
a
−
3
b
,
C
D
→
=
−
2
a
−
8
b
\quad\overrightarrow{CD} = -2\mathbf{a} - 8\mathbf{b}
C
D
=
−
2
a
−
8
b
.
(a)
Find
A
D
→
\overrightarrow{AD}
A
D
in terms of
a
\mathbf{a}
a
and
b
\mathbf{b}
b
. Give your answer in its simplest form.
(b)
Show that
A
D
AD
A
D
is parallel to
B
C
BC
B
C
.
(c)
Aisha says that
A
B
C
D
ABCD
A
B
C
D
is a parallelogram. Aisha is wrong. Explain why.
●●●●
●
Level 4
5 marks
Start
→