Vector Operations

A vector is an arrow carrying both a direction and a size. Join two arrows and the diagonal of the parallelogram is exactly their sum, while multiplying an arrow by a number (a scalar) keeps its direction but stretches, shrinks, or flips it. The neat part is that all of this comes down to adding and multiplying the x- and y-components separately. Drag the components of a and b with the sliders to watch the sum vector appear, then change k to see the arrow stretch and flip.

Vector Addition

a_x3

a_y2

b_x1

b_y3

👀 Intuition for Vector Addition

①Red arrow a and blue arrow b start from origin O

②Form a parallelogram from the two arrowheads

③The diagonal is the sum vector a+b (gold)

④Just add components: (a_x+b_x, a_y+b_y)

Scalar Multiplication

Scalar k1.5

↔️ What is Scalar Multiplication?

①k > 1: extends in the same direction

②0 < k < 1: shrinks in the same direction

③k < 0: flips to the opposite direction

④k = 0: becomes the zero vector

Vector Operation Laws

Vector Addition (Components)

a + b = (a_x + b_x, a_y + b_y)

Add corresponding components

Scalar Multiplication (Components)

ka = (ka_x, ka_y)

Multiply each component by k

📐 Operation Laws

①Commutative: a + b = b + a

②Associative: (a + b) + c = a + (b + c)

③Distributive: k(a + b) = ka + kb

④Distributive: (k + m)a = ka + ma

Position Vectors and Internal/External Division

Position Vector of Internal Division

p = na + mbm + n

Point P that divides segment AB in ratio m:n

Position Vector of Midpoint

p = a + b2

Special case m = n = 1

🎯 Mnemonic for Division Formula

①m:n internal division → cross the opposite ratios

②Denominator is always m+n

③Midpoint is simply the average

④External division: denominator becomes m−n (mind the sign)

Work It Out

Example 1

Given a=(2, 3) and b=(−1, 4), find a+b and 2a−b.

Addition is done component by component.

a + b = (2+(−1), 3+4) = (1, 7)

Scalar multiplication then subtraction is also component-wise.

2a − b = (4−(−1), 6−4) = (5, 2)

▸ a+b=(1, 7), 2a−b=(5, 2)

Vector operations work on the x-components and y-components separately.

Example 2

Given a=(3, −4), find the magnitude |a| and the unit vector in the direction of a.

The magnitude comes from the Pythagorean theorem.

| a | = √(3² + (−4)²) = √25 = 5

The unit vector is a divided by its own magnitude.

a / | a | = (35, −45)

▸ |a|=5, unit vector=(3/5, −4/5)

A unit vector keeps the same direction but has magnitude 1.

Summary

Vector Magnitude

| a | = √(a_x² + a_y²)

Magnitude (length) by Pythagorean theorem

2023 KICE mock exam Math (Geometry) type, adapted

For two vectors a=(1, 2) and b=(3, k), if a + b = (4, 5), what is the value of k?

①1

②2

③3

④4

⑤5

▸ ③ 3

Addition is component-wise, so compare the y-components.

y-component: 2+k=5.

2 + k = 5 ⇒ k = 3

🎯 Exam Points

①Vector addition: add components

②Scalar multiplication changes direction (sign) and magnitude (absolute value)

③Parallel condition: a = kb (some nonzero scalar k exists)

④Memorize the cross order in the division formula

⑤Unit vector: a / |a| (magnitude 1)

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Hyperbola

Inner Product

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