Which of the following expressions represent vectors ?
(Zero or more options can be correct)

For example, choice (a) should be True.

For example, choice (b) should be False.

For example, choice (c) should be False.

For example, choice (d) should be False.

For example, choice (e) should be True.

For example, choice (f) should be True.

*There is at least one mistake.*

For example, choice (a) should be True.

*There is at least one mistake.*

For example, choice (b) should be False.

This is the length, or
magnitude, of the vector $\stackrel{\u20d7}{AB}$.

*There is at least one mistake.*

For example, choice (c) should be False.

This is the length, or
magnitude, of the vector $\mathbf{v}$.

*There is at least one mistake.*

For example, choice (d) should be False.

Although
$\stackrel{\u20d7}{AB}$ is a vector,
$\left|\mathbf{v}\right|$ represents the length,
or magnitude, of $\mathbf{v}$.

*There is at least one mistake.*

For example, choice (e) should be True.

This is the vector pointing in the opposite direction to
$\stackrel{\u20d7}{PQ}$.

*There is at least one mistake.*

For example, choice (f) should be True.

both
$\mathbf{u}$ and
$\stackrel{\u20d7}{AB}$ are
vectors, so their sum is also a vector.

*Correct!*

*True**False*This is the length, or magnitude, of the vector $\stackrel{\u20d7}{AB}$.*False*This is the length, or magnitude, of the vector $\mathbf{v}$.*False*Although $\stackrel{\u20d7}{AB}$ is a vector, $\left|\mathbf{v}\right|$ represents the length, or magnitude, of $\mathbf{v}$.*True*This is the vector pointing in the opposite direction to $\stackrel{\u20d7}{PQ}$.*True*both $\mathbf{u}$ and $\stackrel{\u20d7}{AB}$ are vectors, so their sum is also a vector.

In the notation used in Math1002, which of the following expressions represent
vectors ?
(Zero or more options can be correct)

For example, choice (a) should be True.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be False.

For example, choice (e) should be True.

*There is at least one mistake.*

For example, choice (a) should be True.

Minus a vector is a vector.

*There is at least one mistake.*

For example, choice (b) should be True.

The
sum of two vectors is a vector.

*There is at least one mistake.*

For example, choice (c) should be False.

As
$\mathbf{u}$ is a vector and
$\left|\mathbf{v}\right|$ is a number (namely,m
the length of $\mathbf{v}$),
this expression does not even make sense.

*There is at least one mistake.*

For example, choice (d) should be False.

As
$\stackrel{\u20d7}{AB}$ is a vector, this is
minus the length of $\stackrel{\u20d7}{AB}$.

*There is at least one mistake.*

For example, choice (e) should be True.

Any
linear combination of vectors is again a vector.

*Correct!*

*True*Minus a vector is a vector.*True*The sum of two vectors is a vector.*False*As $\mathbf{u}$ is a vector and $\left|\mathbf{v}\right|$ is a number (namely,m the length of $\mathbf{v}$), this expression does not even make sense.*False*As $\stackrel{\u20d7}{AB}$ is a vector, this is minus the length of $\stackrel{\u20d7}{AB}$.*True*Any linear combination of vectors is again a vector.

How many different vectors are drawn here ?

*Correct!*

Vectors are equal when they have the same direction and length, their position in
space does not matter.

*Incorrect.*

*Please try again.*

Remember that a vector is specified by its direction and magnitude, so that the two
arrows of equal length pointing to the west represent the same vector, while the three
arrows of equal length pointing to the north-east also represent the same
vector.

How many different vectors are drawn here ?

*Correct!*

All of the vectors are different. However, some of these vectors are multiples of each
other; for example,

*Incorrect.*

*Please try again.*

Vectors are equal when they have the same direction and length, their position in
space does not matter.

Express the vector $\mathbf{u}$
in terms of $\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b},\phantom{\rule{0.3em}{0ex}}\mathbf{c}$.
Exactly one option must be correct)

*Choice (a) is incorrect*

Trace out the vector $\mathbf{u}$
starting at the tail and moving along the vectors
$\mathbf{a}$,
$\mathbf{b}$ and
$\mathbf{c}$.

*Choice (b) is incorrect*

Trace out
the vector $\mathbf{u}$
starting at the tail and moving along the vectors
$\mathbf{a}$,
$\mathbf{b}$ and
$\mathbf{c}$.

*Choice (c) is incorrect*

Trace out
the vector $\mathbf{u}$
starting at the tail and moving along the vectors
$\mathbf{a}$,
$\mathbf{b}$ and
$\mathbf{c}$.

*Choice (d) is correct!*

Starting at
the tail of $\mathbf{u}$,
we see that $\mathbf{u}=-\mathbf{a}-\mathbf{b}-\mathbf{c}$.

Express the vector $\mathbf{u}$
in terms of $\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b},\phantom{\rule{0.3em}{0ex}}\mathbf{c}$.
Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is correct!*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Find non-zero scalars $\alpha $,
$\beta $ such that for
all vectors $\mathbf{a}$
and $\mathbf{b}$,
$$\alpha \left(\mathbf{a}+2\mathbf{b}\right)-\beta \mathbf{a}+\left(4\mathbf{b}-\mathbf{a}\right)=\mathbf{0}.$$
Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is correct!*

If the vector equation is simplified, we get
$$\left(\alpha -\beta -1\right)\mathbf{a}+\left(2\alpha +4\right)\mathbf{b}=\mathbf{0}.$$
Since this holds for all $\mathbf{a}$
and $\mathbf{b}$, it will
hold when $\mathbf{a}$ and
$\mathbf{b}$ are set equal to
$\mathbf{0}$ in turn. This then
gives two conditions $\alpha -\beta -1=0$
and $2\alpha +4=0$, whose
solution is $\alpha =-2,\phantom{\rule{0.3em}{0ex}}\beta =-3.$

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Find non-zero scalars $\alpha $,
$\beta $ such that for
all vectors $\mathbf{a}$
and $\mathbf{b}$,
$$\alpha \left(2\mathbf{a}-\mathbf{b}\right)-\beta \left(\mathbf{a}+2\mathbf{b}\right)=4\mathbf{a}-\mathbf{b}.$$
Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

We have two equations
$$2\alpha -\beta =4,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}-\alpha -2\beta =-1.$$
The substitution $\alpha =1-2\beta $
implies $2\left(1-2\beta \right)-\beta =5$
and so $\beta =-\frac{2}{5}$,
$\alpha =1+\frac{4}{5}=\frac{9}{5}$.

*Choice (d) is incorrect*

The two vectors $\mathbf{a}$
and $\mathbf{b}$ are
perpendicular. If $\mathbf{a}$
has magnitude $8$
and $\mathbf{b}$ has
magnitude $3$
what is $|\mathbf{a}-2\mathbf{b}|$?

*Correct!*

The vectors $\mathbf{a}$,
$-2\mathbf{b}$, and
$\mathbf{a}-2\mathbf{b}$
form the sides of a right-angled triangle, with sides of length
$8$,
$6$ and hypotenuse of
length $|\mathbf{a}-2\mathbf{b}|$. Therefore by
Pythagoras’ Theorem, $|\mathbf{a}-2\mathbf{b}|=\sqrt{{8}^{2}+{6}^{2}}=10.$

*Incorrect.*

*Please try again.*

Try drawing a diagram!

In which of the following cases is the length of
$\mathbf{a}+\mathbf{b}$ strictly smaller
than the length of $\mathbf{a}-\mathbf{b}$
?
(Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be False.

For example, choice (c) should be False.

For example, choice (d) should be True.

*There is at least one mistake.*

For example, choice (a) should be False.

The parallelogram rule for vector addition shows that when
$\mathbf{a}$ and
$\mathbf{b}$
are placed tail to tail, the diagonals of the parallelogram are
$\mathbf{a}+\mathbf{b}$ and
$\mathbf{a}-\mathbf{b}$. Hence, in
this case, $|\mathbf{a}+\mathbf{b}|>|\mathbf{a}-\mathbf{b}|$.

*There is at least one mistake.*

For example, choice (b) should be False.

The parallelogram rule for vector addition shows that when
$\mathbf{a}$ and
$\mathbf{b}$
are placed tail to tail, the diagonals of the parallelogram are
$\mathbf{a}+\mathbf{b}$ and
$\mathbf{a}-\mathbf{b}$. Hence, in
this case, $|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|$.

*There is at least one mistake.*

For example, choice (c) should be False.

The parallelogram rule for vector addition shows that when
$\mathbf{a}$ and
$\mathbf{b}$
are placed tail to tail, the diagonals of the parallelogram are
$\mathbf{a}+\mathbf{b}$ and
$\mathbf{a}-\mathbf{b}$. Hence, in
this case, $|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|$.

*There is at least one mistake.*

For example, choice (d) should be True.

The parallelogram rule for vector addition shows that when
$\mathbf{a}$ and
$\mathbf{b}$
are placed tail to tail, the diagonals of the parallelogram are
$\mathbf{a}+\mathbf{b}$ and
$\mathbf{a}-\mathbf{b}$. Hence, in
this case, $|\mathbf{a}+\mathbf{b}|<|\mathbf{a}-\mathbf{b}|$.

*Correct!*

*False*The parallelogram rule for vector addition shows that when $\mathbf{a}$ and $\mathbf{b}$ are placed tail to tail, the diagonals of the parallelogram are $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$. Hence, in this case, $|\mathbf{a}+\mathbf{b}|>|\mathbf{a}-\mathbf{b}|$.*False*The parallelogram rule for vector addition shows that when $\mathbf{a}$ and $\mathbf{b}$ are placed tail to tail, the diagonals of the parallelogram are $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$. Hence, in this case, $|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|$.*False*The parallelogram rule for vector addition shows that when $\mathbf{a}$ and $\mathbf{b}$ are placed tail to tail, the diagonals of the parallelogram are $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$. Hence, in this case, $|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|$.*True*The parallelogram rule for vector addition shows that when $\mathbf{a}$ and $\mathbf{b}$ are placed tail to tail, the diagonals of the parallelogram are $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$. Hence, in this case, $|\mathbf{a}+\mathbf{b}|<|\mathbf{a}-\mathbf{b}|$.