Vectors flashcards
9 cards on Vectors: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 9
Vectors
What a vector is
What is a vector, and how is one different from a scalar?
Answer it in your head, then turn the card over.
What a vector is
Key points
- A vector has a size (its magnitude) and a direction: 5 km north, not just 5 km.
- A scalar has a size only, such as a mass or a speed.
- It is written as a bold letter, a, underlined in handwriting, or as AB with an arrow over it.
- Two vectors are equal when they have the same size and the same direction, wherever they are drawn.
Watch out
AB and BA are not the same vector: same length, opposite directions.
Vectors
Column vectors
How does a column vector describe a movement?
Answer it in your head, then turn the card over.
Column vectors
Key points
- The top number is the move across, the bottom the move up or down: (3 over −2) is 3 right, 2 down.
- Moves left and down are negative, as on the x and y axes.
- Write the two numbers one above the other in tall brackets, with no line between them.
- A vector says where you end up compared with where you started, whatever the route.
Start at (2, 3), move by (−4 over 1)
4 left and 1 up: (2 − 4, 3 + 1)
You end at (−2, 4).
Watch out
Mind the signs: 2 left and 3 down is (−2 over −3), not (2 over 3).
Vectors
Adding and subtracting vectors
How do you add and subtract column vectors, and what does adding mean as a journey?
Answer it in your head, then turn the card over.
Adding and subtracting vectors
Key points
- Add or subtract the matching numbers: tops with tops, bottoms with bottoms.
- Adding is one journey followed by the other; drawn end to end, the sum is the third side of the triangle.
- a − b is the same as a + (−b). The order of adding does not matter: a + b = b + a.
(4 over −1) and (−2 over 5)
Sum: 4 + (−2) = 2 and −1 + 5 = 4, so (2 over 4)
Difference: 4 − (−2) = 6 and −1 − 5 = −6, so (6 over −6)
Watch out
Subtracting a negative adds: 4 − (−2) is 6, not 2.
Vectors
Multiplying a vector by a number
What happens when you multiply a vector by a number, including a negative one?
Answer it in your head, then turn the card over.
Multiplying a vector by a number
Key points
- Multiply both numbers in the column vector, the top and the bottom.
- 3a points the same way as a and is three times as long; ½a is half as long.
- −a is the same length as a but points exactly the opposite way.
- Any number multiple of a is parallel to a.
a = (2 over −4)
3a: 3 × 2 = 6 and 3 × (−4) = −12, so (6 over −12)
−a = (−2 over 4)
Watch out
Multiply the bottom number too: 3 × (2 over −4) is (6 over −12), not (6 over −4).
Vectors
Working out expressions like 2a − 3b
Given two column vectors a and b, how do you work out an expression such as 2a − 3b?
Answer it in your head, then turn the card over.
Working out expressions like 2a − 3b
Key points
- Multiply each vector by its number first.
- Then add or subtract the matching numbers.
- Take care with signs, especially when subtracting a negative number.
a = (3 over 1), b = (−1 over 2)
2a = (6 over 2) and 3b = (−3 over 6)
Top: 6 − (−3) = 9. Bottom: 2 − 6 = −4. So 2a − 3b = (9 over −4)
Watch out
Subtract all of 3b, sign included: the top is 6 − (−3) = 9, not 6 − 3 = 3.
Vectors
HigherFinding a vector by a route
How do you write the vector between two points in terms of given vectors a and b?
Answer it in your head, then turn the card over.
Finding a vector by a route
Key points
- Trace a route between the two points through points whose vectors you know.
- Add the vector for each step; a step against its arrow gets a minus sign.
- Every route between the same two points gives the same vector, so pick an easy one.
- Simplify like algebra: collect the a terms, then the b terms.
OA = a and OB = b; find AB
A to O is against the arrow: −a
AB = AO + OB = −a + b = b − a
Watch out
Going against an arrow flips the sign: if AB = a, then BA = −a, not a.
Vectors
HigherPoints that divide a line
How do you find the vector to a midpoint, or to a point that divides a line in a ratio?
Answer it in your head, then turn the card over.
Points that divide a line
Key points
- The midpoint M of AB: AM = ½AB.
- P divides AB in the ratio m : n: AP is of AB, and PB is of AB.
- Find AB by a route first, then take the fraction of it.
OA = a, OB = b; AP : PB = 1 : 3
AB = b − a, split into 1 + 3 = 4 equal parts
AP = of AB = ¼(b − a)
OP = OA + AP = a + ¼(b − a) = ¾a + ¼b
Watch out
A ratio of 1 : 3 makes quarters, not thirds: AP is ¼ of AB, not ⅓.
Vectors
HigherShowing vectors are parallel
How can you tell from their expressions that two vectors are parallel?
Answer it in your head, then turn the card over.
Showing vectors are parallel
Key points
- Two vectors are parallel when one is a number multiple of the other.
- Take out a common factor to spot it: 6a − 3b = 3(2a − b), so it is parallel to 2a − b.
- The number outside the bracket says how many times as long it is.
- A negative multiple is parallel too, but points the other way.
Is 4a + 10b parallel to 2a + 5b?
4 ÷ 2 = 2 and 10 ÷ 5 = 2
4a + 10b = 2(2a + 5b): parallel, and twice as long.
Watch out
Both parts need the same multiple: 4a + 6b is not parallel to 2a + 5b, since 4 ÷ 2 and 6 ÷ 5 differ.
Vectors
HigherShowing points lie on a straight line
How do you prove with vectors that three points A, B and C lie on a straight line?
Answer it in your head, then turn the card over.
Showing points lie on a straight line
Key points
- Find AB and BC in terms of the given vectors.
- Show that one is a multiple of the other, so they are parallel.
- Say that they share the point B.
- Write both facts in words: parallel and a shared point mean one straight line.
AB = 3a − 6b and BC = a − 2b
3(a − 2b) = 3a − 6b, so AB = 3BC: AB and BC are parallel.
They share the point B, so A, B and C lie on a straight line.
Watch out
Parallel alone is not enough: two parallel lines can be separate, like railway tracks. The shared point proves it.
Print these cards
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One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
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