What two things must a drawing of a vector show?

Its size , shown by the length of the line, and its direction , shown by an arrow on that line. A vector has both of these, which is what distinguishes it from a scalar.

True or False? A vector must be drawn starting from one particular point on the grid.

False. A vector can be drawn anywhere on the grid, provided it has the right length and the right direction. It describes a movement rather than a position, so where you start drawing it does not change which vector it is.

Define the modulus of a vector.

The modulus of a vector is its length , also called its magnitude . It is always positive , and the vector's direction plays no part in it.

What does the modulus of a velocity vector represent?

It represents the speed , which is the size of the velocity with its direction stripped away. For a force vector the modulus is instead the strength of the force, measured in newtons.

Define a unit vector .

A unit vector is a vector whose modulus is 1 . It carries a direction but no size beyond that, so it is the natural way to describe a direction on its own.

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Vectors (Edexcel IGCSE Maths B): Flashcards

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FrontIntroduction to Column Vectors
What does the column vector $(\begin{matrix} 6 \\ 3 \end{matrix})$ mean?

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What does the column vector $(\begin{matrix} 6 \\ 3 \end{matrix})$ mean?
It describes a movement of 6 units to the right and 3 units up.
A column vector used this way is also called a translation vector, because it says how to get from one point to another.
How do you add or subtract two column vectors?
Work on the top numbers and the bottom numbers separately, keeping the two apart throughout.
So $(\begin{matrix} 5 \\ 2 \end{matrix}) + (\begin{matrix} 3 \\ - 1 \end{matrix}) = (\begin{matrix} 8 \\ 1 \end{matrix})$ .
Fill in the two missing numbers.
$3 (\begin{matrix} 2 \\ - 1 \end{matrix}) = (\begin{matrix} ______ \\ ______ \end{matrix})$
The completed calculation is:
$3 (\begin{matrix} 2 \\ - 1 \end{matrix}) = (\begin{matrix} 6 \\ - 3 \end{matrix})$
A scalar is an ordinary number with no direction attached, and it multiplies both components.
How do you write $2 (\begin{matrix} 5 \\ 2 \end{matrix}) + 5 (\begin{matrix} 3 \\ - 1 \end{matrix})$ as a single column vector?
Multiply each vector by the scalar in front of it first, which gives $(\begin{matrix} 10 \\ 4 \end{matrix}) + (\begin{matrix} 15 \\ - 5 \end{matrix})$ .
Then add those results component by component to get $(\begin{matrix} 25 \\ - 1 \end{matrix})$ .
Given that $(\begin{matrix} 2 p - 6 \\ 9 \end{matrix}) = (\begin{matrix} 4 \\ q \end{matrix})$ , how do you find $p$ and $q$ ?
Two column vectors are equal only when both components match, so each pair can be equated separately.
The top components give $2 p - 6 = 4$ and so $p = 5$ , while the bottom components give $q = 9$ straight away.
True or False?
A column vector can be treated as a $2 \times 1$ matrix.
True.
A column vector has 2 rows and 1 column, which is exactly what a $2 \times 1$ matrix is.
That is why a $2 \times 2$ matrix can be multiplied by one, and it is what makes matrix transformations possible.
What two things must a drawing of a vector show?
Its size, shown by the length of the line, and its direction, shown by an arrow on that line.
A vector has both of these, which is what distinguishes it from a scalar.
Fill in the two missing words about writing vectors.
A vector is written in $______$ when it is typed, and is $______$ when it is written by hand.
The completed sentence is:
A vector is written in bold when it is typed, and is underlined when it is written by hand.
The two conventions mean the same thing, and handwriting uses the underline simply because you cannot write in bold with a pen.
What is the difference between $\vec{A B}$ and $\vec{B A}$ ?
They have the same length but point in opposite directions.
$\vec{A B}$ starts at $A$ and points towards $B$ , while $\vec{B A}$ starts at $B$ and points towards $A$ .
True or False?
A vector must be drawn starting from one particular point on the grid.
False.
A vector can be drawn anywhere on the grid, provided it has the right length and the right direction.
It describes a movement rather than a position, so where you start drawing it does not change which vector it is.
What does multiplying a vector by $- 2$ do to its arrow?
It reverses the direction and makes the arrow twice as long.
A negative scalar always reverses the direction, and it is the number after the minus sign that gives the change in length.
How do you draw the vector $a + b$ ?
Draw $a$ , then draw $b$ starting from the point where $a$ ends.
The sum is the line from the start of $a$ to the end of $b$ , and for $a - b$ you draw $- b$ second instead.
Why is $\vec{A B} + \vec{B C} + \vec{C A}$ the zero vector?
Because following those three vectors in turn takes you from $A$ all the way back to $A$ .
A journey that returns to where it started has no overall movement, so the result is $(\begin{matrix} 0 \\ 0 \end{matrix})$ .
Define the modulus of a vector.
The modulus of a vector is its length, also called its magnitude.
It is always positive, and the vector's direction plays no part in it.
Complete the formula for the modulus of a vector with components $x$ and $y$ .
$|a| = \sqrt{x^{2} +______}$
The completed formula is:
$|a| = \sqrt{x^{2} + y^{2}}$
This is Pythagoras' theorem: the two components are the shorter sides of a right-angled triangle and the vector itself is the hypotenuse.
Points $A (- 3, 5)$ and $B (7, 1)$ are given. What is $|\vec{A B}|$ ?
First $\vec{A B} = (\begin{matrix} 10 \\ - 4 \end{matrix})$ , found by subtracting the coordinates of $A$ from those of $B$ .
Then $|\vec{A B}| = \sqrt{10^{2} + {(- 4)}^{2}} = \sqrt{116} = 2 \sqrt{29}$ .
True or False?
$|\vec{B A}|$ and $|\vec{A B}|$ are equal.
True.
Both vectors have the same length, and the modulus ignores direction entirely.
Reversing a vector changes the signs of its components, but squaring them in the formula removes those signs again.
What does the modulus of a velocity vector represent?
It represents the speed, which is the size of the velocity with its direction stripped away.
For a force vector the modulus is instead the strength of the force, measured in newtons.
A vector $\vec{C D}$ has three times the modulus of $\vec{A B} = (\begin{matrix} 10 \\ - 4 \end{matrix})$ . Give one possible $\vec{C D}$ .
Multiplying both components by 3 gives $(\begin{matrix} 30 \\ - 12 \end{matrix})$ , which is three times as long.
Multiplying by $- 3$ gives $(\begin{matrix} - 30 \\ 12 \end{matrix})$ , which is also three times as long but points the opposite way, so it works just as well.
True or False?
$\vec{A B} = \vec{O A} - \vec{O B}$
False.
$\vec{A B} \neq \vec{O A} - \vec{O B}$
The correct equation is $\vec{A B} = \vec{O B} - \vec{O A}$ .
How do you show that $(\begin{matrix} 9 \\ - 3 \end{matrix})$ and $(\begin{matrix} - 6 \\ 2 \end{matrix})$ are parallel?
Factorise each of them: $(\begin{matrix} 9 \\ - 3 \end{matrix}) = 3 (\begin{matrix} 3 \\ - 1 \end{matrix})$ and $(\begin{matrix} - 6 \\ 2 \end{matrix}) = - 2 (\begin{matrix} 3 \\ - 1 \end{matrix})$ .
Both turn out to be multiples of the same vector, and that is what shows they are parallel.
If $a$ and $b$ are parallel, what equation can you write down?
Introduce a scalar $k$ and write $a = k b$ .
That turns a statement about direction into an equation you can actually solve for unknown components.
True or False?
Two vectors pointing in exactly opposite directions are still parallel.
True.
One of them is a negative scalar multiple of the other, and that still counts as parallel.
For example $(\begin{matrix} 2 \\ - 4 \end{matrix})$ and $(\begin{matrix} - 3 \\ 6 \end{matrix})$ are parallel, because the second is $- 1.5$ times the first.
Define a unit vector.
A unit vector is a vector whose modulus is 1.
It carries a direction but no size beyond that, so it is the natural way to describe a direction on its own.
Complete the rule for a unit vector in the direction of $a$ .
$\frac{a}{______}$
The completed rule is:
$\frac{a}{|a|}$
Dividing a vector by its own modulus scales its length to exactly 1 while leaving its direction untouched.
What is the unit vector in the direction of $(\begin{matrix} 3 \\ - 4 \end{matrix})$ ?
Its modulus is $\sqrt{3^{2} + {(- 4)}^{2}} = 5$ , so divide each component by 5.
That gives $(\begin{matrix} \frac{3}{5} \\ - \frac{4}{5} \end{matrix})$ .
Why can a unit vector have surds in its components?
Because the modulus you divide by is often not a whole number, and dividing by a surd leaves one behind.
The unit vector in the direction of $(\begin{matrix} - 2 \\ 5 \end{matrix})$ is $(\begin{matrix} - \frac{2}{\sqrt{29}} \\ \frac{5}{\sqrt{29}} \end{matrix})$ , and that is an exact answer rather than an unfinished one.
True or False?
$\vec{A B} + \vec{B C} = \vec{A C}$
True.
$\vec{A B} + \vec{B C} = \vec{A C}$
This means if you start at A then go to B and then go to C, this is the same vector as starting at A and going to C.
How else could you write the vector $- \vec{A B}$ ?
The vector $- \vec{A B}$ can also be written as $\vec{B A}$ .
True or False?
If two vectors are parallel, then they are scalar multiples.
True.
If two vectors are parallel, then they are scalar multiples.
E.g. the vectors $3 a$ and $12 a$ are parallel.
True or False?
If the point P divides the line AB in the ratio 1 : 3, then $\vec{A P} = \frac{1}{3} \vec{A B}$ .
False.
If the point P divides the line AB in the ratio 1 : 3, then $\vec{A P} \neq \frac{1}{3} \vec{A B}$ .
P divides the line into 4 equal parts, with 1 part on one side, and 3 parts on the other side.
The correct equation is $\vec{A P} = \frac{1}{4} \vec{A B}$ .
If the vectors $\vec{A B}$ and $\vec{A C}$ are parallel, what does this tell you about the three points, $A$ , $B$ and $C$ ?
If the vectors $\vec{A B}$ and $\vec{A C}$ are parallel, then the three points $A$ , $B$ and $C$ all lie on the same straight line.
Points A and B have position vectors $a$ and $b$ , respectively.
Write a formula for the position vector of the midpoint of A and B in terms of $a$ and $b$ .
Points A and B have position vectors $a$ and $b$ , respectively.
A formula for the position vector of the midpoint of A and B is $\frac{1}{2} (a + b)$ .

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