A vector has both a size and a direction, so it is drawn as an arrow.
Start two vectors at the origin and complete a parallelogram, and its diagonal is their sum; multiply a vector by a real number k and its length becomes |k| times as long while its direction stays the same or flips.
All of this can be computed separately on the x-components and the y-components, so what you see in the picture always matches what the components give.
Change a and b with the component sliders to see the sum and its components change together, and move k to stretch and shrink the arrow and flip it when k is negative.
Adding Vectors
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Two vectors a and b and the diagonal a + b of their parallelogram
3
2
1
3
👀 What to read in the picture
①The red arrow a and the blue arrow b start at the origin O
②The dashed segments go b further from the tip of a and a further from the tip of b, meeting at the fourth corner of the parallelogram
③The gold diagonal from the origin to that corner is the sum a + b
④As the top band shows, adding x-components to x-components and y-components to y-components gives the same result
Scalar Multiples
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kv, the vector v = (3, 2) multiplied by a real number k
1.5
↔️ What multiplying by k does
①If k > 1, the vector stretches in the same direction
②If 0 < k < 1, it shrinks in the same direction
③If k < 0, its direction flips and its length becomes |k| times as long
④If k = 0, it becomes the zero vector
⑤Two nonzero vectors a and b are parallel exactly when a = kb for some real number k
Rules for Vector Operations
Addition (components)
a + b = (ax + bx, ay + by)
Add component by component
Scalar multiple (components)
ka = (kax, kay)
Multiply each component by k
📐 Laws of operation
①Commutative law: a + b = b + a
②Associative law: (a + b) + c = a + (b + c)
③Distributive law: k(a + b) = ka + kb
④Distributive law: (k + m)a = ka + ma
⑤Dividing a nonzero vector a by its length |a| gives the unit vector in the same direction
Position Vectors and Dividing a Segment
Position vector of an internal division point
p = na + mbm + n
The point P dividing segment AB internally in the ratio m : n
Position vector of the midpoint
p = a + b2
The case m = n = 1
🎯 Remembering the division formula
①For internal division in the ratio m : n, a gets the factor n and b gets m, crosswise
②The denominator is m + n
③The midpoint is the average of the two position vectors
④For external division in the ratio m : n, the denominator is m − n and the sign of the coefficient of a changes
Worked Examples
Example 1
If a = (2, 3) and b = (−1, 4), find a + b and 2a − b.
1
Add component by component.
a + b = (2 + (−1), 3 + 4) = (1, 7)
2
Multiplying by a scalar and then subtracting also works component by component.
2a − b = (4 − (−1), 6 − 4) = (5, 2)
▸ a + b = (1, 7), 2a − b = (5, 2)
Vector operations can be done separately on the x-components and the y-components.
Example 2
If a = (3, −4), find its length |a| and the unit vector in the direction of a.
1
Find the length with the Pythagorean theorem.
| a | = √(3² + (−4)²) = √25 = 5
2
The unit vector is a divided by its own length.
a / | a | = (35, −45)
▸ |a| = 5, unit vector (3/5, −4/5)
A unit vector points the same way and has length 1.
Summary
Length of a vector
| a | = √(ax² + ay²)
Use the Pythagorean theorem to find the length
2023 KICE mock exam Math (Geometry) type, adapted
For the vectors a = (1, 2) and b = (3, k), if a + b = (4, 5), what is k?
①1
②2
③3
④4
⑤5
▸ ③ 3
1
Addition is done component by component, so compare the y-components.
2
The y-components give 2 + k = 5.
2 + k = 5 ⇒ k = 3
🎯 Exam Points
①Vectors are added component by component
②Multiplying by k changes the direction through the sign of k and the length through |k|
③The parallel condition is a = kb for some nonzero real k
④In the division formula, the closer P is to A, the larger the coefficient of a