A quantity with both magnitude and direction , representing a movement of a given size in a given direction. It is the direction that separates a vector from a scalar, which carries size alone.

Why is a vector often described as a journey?

Because it records only the movement from one point to another, and not where that movement began. The same vector can be drawn anywhere on a diagram and it is still the same vector.

True or False? Two vectors drawn in different places on a diagram can still be equal.

True. Vectors are equal if they have the same magnitude and direction , wherever they happen to be drawn. Opposite sides of a parallelogram are the standard example: same length, same direction, different positions.

Define resultant vector.

The single vector produced by adding two or more vectors together. It describes the overall movement from the start of the first to the end of the last, whatever route was taken in between.

How do you add two vectors?

Place them nose to tail , so that the second begins where the first finishes, and the resultant runs from the very start to the very end. In component form you simply add the corresponding components.

How does a position vector differ from a displacement vector?

A position vector is measured from the origin, so it fixes where a point actually is. A displacement vector runs between any two points and gives only the direction and distance from one to the other.

What does a vector proof of a geometric fact look like?

Express every vector you need in terms of two known ones, then show the relationship you want algebraically. Choosing which two to build everything from is the whole difficulty, and a labelled diagram is what makes the choice obvious.

Why use vectors to prove a geometric result at all?

Because a relationship that would otherwise need a construction or an angle chase becomes ordinary algebra. Parallel, collinear and ratio facts all turn into statements about scalar multiples, which are easy to check.

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Vectors in 2D (Edexcel A Level Maths: Pure): Flashcards

Exam code: 9MA0

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FrontBasic Vectors
Define vector.

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Cards in this collection (26)

Define vector.
A quantity with both magnitude and direction, representing a movement of a given size in a given direction.
It is the direction that separates a vector from a scalar, which carries size alone.
What is the difference between a speed of $7$ and a velocity of $7$ ?
Speed is a scalar, so the number is the whole story.
Velocity is a vector and needs a direction too, which is why velocities of $7$ and $- 7$ describe the same speed travelling opposite ways.
How do you write vector notation by hand?
Underline the letter, since bold cannot be written by hand.
A vector printed as $a$ is written $a$ in your own working, and the underlining is not optional.
Why is a vector often described as a journey?
Because it records only the movement from one point to another, and not where that movement began.
The same vector can be drawn anywhere on a diagram and it is still the same vector.
True or False?
Two vectors drawn in different places on a diagram can still be equal.
True.
Vectors are equal if they have the same magnitude and direction, wherever they happen to be drawn.
Opposite sides of a parallelogram are the standard example: same length, same direction, different positions.
How do you find the magnitude of a 2D vector?
By Pythagoras' theorem on its components, so $|x i + y j| = \sqrt{x^{2} + y^{2}}$ .
The two components are the shorter sides of a right-angled triangle and the vector itself is the hypotenuse.
How is the direction of a 2D vector specified?
As an angle measured anticlockwise from the positive $x$ -axis, unless the question says otherwise.
It is found from $\tan θ = \frac{y}{x}$ , using the components as the two sides of a right-angled triangle.
A vector has magnitude $r$ and direction $θ$ . What are its components?
$x = r \cos θ$ and $y = r \sin θ$ , which is the same right-angled triangle read the other way round.
Writing a vector in component form this way is called resolving it.
True or False?
$\tan θ = \frac{y}{x}$ always gives the direction of the vector.
False.
A calculator returns an angle between $- 90 °$ and $90 °$ , so it cannot distinguish a vector from the one pointing in exactly the opposite direction.
A sketch settles which quadrant the vector lies in, and the angle is adjusted to match.
What is the relationship between $a$ and $- a$ ?
The same magnitude, and exactly the opposite direction.
Every component changes sign, so $\vec{B C}$ becoming $\vec{C B}$ reverses the journey without changing its length.
Define resultant vector.
The single vector produced by adding two or more vectors together.
It describes the overall movement from the start of the first to the end of the last, whatever route was taken in between.
How do you add two vectors?
Place them nose to tail, so that the second begins where the first finishes, and the resultant runs from the very start to the very end.
In component form you simply add the corresponding components.
What does subtracting a vector mean?
Adding the negative of it, so $a - b$ means $a + (- b)$ .
On a diagram that is travelling along $b$ backwards.
There is no direct route from $A$ to $B$ on a diagram. How do you find $\vec{A B}$ ?
Take a detour through points you do know, such as $\vec{A B} = \vec{A O} + \vec{O B}$ .
Every route from $A$ to $B$ gives the same resultant, so choose whichever one uses vectors you already have.
Complete the statement:
$\vec{P Q} + \vec{Q P} =______$
The completed statement is:
$\vec{P Q} + \vec{Q P} = 0$
Going from $P$ to $Q$ and straight back again leaves no overall movement, and the zero vector is written as a bold zero.
True or False?
Multiplying a vector by a positive scalar changes its direction.
False.
A positive scalar changes only the size: $3 a$ points the same way as $a$ and is three times as long.
It takes a negative scalar to reverse the direction.
Define position vector.
A vector describing the position of a point relative to the origin, written $\vec{O A}$ or simply $a$ .
Unlike other vectors it is tied to one place, since it must always start at $O$ .
How does a position vector differ from a displacement vector?
A position vector is measured from the origin, so it fixes where a point actually is.
A displacement vector runs between any two points and gives only the direction and distance from one to the other.
Points $A$ and $B$ have position vectors $a$ and $b$ . Complete the rule:
$\vec{A B} =______$
The completed rule is:
$\vec{A B} = b - a$
It comes straight from the detour $\vec{A B} = \vec{A O} + \vec{O B}$ , since $\vec{A O}$ is $- a$ .
How are a point's coordinates related to its position vector?
They are the same numbers: the point $(x, y)$ has position vector $x i + y j$ .
That is what lets a vector question be answered with coordinate geometry, and a coordinate question be answered with vectors.
True or False?
$\vec{A B}$ and $\vec{B A}$ are the same vector.
False.
They share a magnitude but point opposite ways, so $\vec{B A} = - \vec{A B}$ .
In terms of position vectors, one is $b - a$ and the other is $a - b$ .
$P$ divides $A B$ in the ratio $2 : 3$ . How do you find its position vector?
Travel to $A$ , then go the right fraction of the way along $\vec{A B}$ , giving $p = a + \frac{2}{5} (b - a)$ .
The fraction is the first number of the ratio over the total, so $2 : 3$ gives $\frac{2}{5}$ and never $\frac{2}{3}$ .
What is the position vector of the midpoint of $A B$ ?
$\frac{1}{2} (a + b)$ , which is the average of the two position vectors.
It is the ratio result with $1 : 1$ , putting the point half of the way along $\vec{A B}$ from $A$ .
What does a vector proof of a geometric fact look like?
Express every vector you need in terms of two known ones, then show the relationship you want algebraically.
Choosing which two to build everything from is the whole difficulty, and a labelled diagram is what makes the choice obvious.
Why use vectors to prove a geometric result at all?
Because a relationship that would otherwise need a construction or an angle chase becomes ordinary algebra.
Parallel, collinear and ratio facts all turn into statements about scalar multiples, which are easy to check.
True or False?
If $P$ divides $A B$ in the ratio $3 : 1$ , then $P$ is three quarters of the way from $A$ to $B$ .
True.
The ratio splits $A B$ into $3 + 1 = 4$ equal parts, and $P$ sits at the end of the third.
So $p = a + \frac{3}{4} (b - a)$ .

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