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.

A LevelMathsCambridge (CIE)Pure 3FlashcardsVectorsVectors in 2 Dimensions

Vectors in 2 Dimensions (Cambridge (CIE) A Level Maths: Pure 3): Flashcards

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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 $a - b$ .
True or False?
If $\vec{O B} = \vec{O A} + \vec{O C}$ , then $O A B C$ is a parallelogram.
True.
The two statements are equivalent, so either one proves the other.
Adding $\vec{O A}$ and $\vec{O C}$ nose to tail lands you at $B$ , which is exactly what it means for $O A$ and $C B$ to be parallel and the same length.
How do you show that two vectors are parallel?
Show that one is a scalar multiple of the other.
In a figure, simplify both vectors in terms of the same two base vectors and then check whether one is a multiple of the other, as with $\vec{C D} = 2 \vec{A B}$ .
$M$ is the midpoint of $A B$ . Complete its position vector in terms of those of $A$ and $B$ :
$\vec{O M} =______$
The completed result is:
$\vec{O M} = \frac{1}{2} (\vec{O A} + \vec{O B})$
You reach $M$ by going to $A$ and then half way along $\vec{A B}$ , and $\vec{O A} + \frac{1}{2} (\vec{O B} - \vec{O A})$ simplifies to the same thing.
Why is showing that $\vec{P Q}$ and $\vec{R S}$ are parallel not enough to prove that $P$ , $Q$ , $R$ and $S$ are collinear?
Because parallel vectors can sit anywhere on the plane without ever meeting.
The two vectors have to share a point as well, which is why collinearity is shown with a pair such as $\vec{P Q}$ and $\vec{P R}$ , both starting at $P$ .
How do you find the position vector of a missing fourth vertex of a parallelogram?
Use the fact that opposite sides of a parallelogram are represented by the same vector.
In parallelogram $P Q R S$ that gives $\vec{S R} = \vec{P Q}$ , so the missing vertex is found by adding a known side vector to a known vertex.

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