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Vector Components English Study Handout

An English-first handout on vector components, vector operations, division points, and space vectors with short explanations and practice answers.

April 23, 2026 修考 10 min read

Vector Components Study Handout

1. Position Vectors and Components in the Plane

A position vector shows the location of a point from the origin. If point A has coordinates (a1, a2), then the vector OA can be written as a = (a1, a2).

In the xy-plane, we use two basic vectors:

  • e1 = (1, 0)
  • e2 = (0, 1)

So every plane vector can be written in basic-vector form:

a = (a1, a2) = a1e1 + a2e2

This means the first number is the x-part, and the second number is the y-part.

Key words

  • position vector: a vector from the origin to a point
  • component: each coordinate of a vector
  • basic vector: the standard unit direction, such as e1 or e2

Examples

  1. Write a = (2, 3) in basic-vector form.
  2. If e1 = (1, 0) and e2 = (0, 1), what is (4, -1) in basic-vector form?
  3. What does the second component of (5, 7) mean?

Answers

  1. a = 2e1 + 3e2
  2. 4e1 - e2
  3. It is the y-component, so it shows the vertical part.

2. Vector From One Point to Another and Component Operations

If P(p1, p2) and Q(q1, q2), then

PQ = (q1 - p1, q2 - p2)

This is a very important rule. We subtract the starting point from the ending point.

Vector operations can also be done component by component:

  • equality: (a1, a2) = (b1, b2) means a1 = b1 and a2 = b2
  • addition: (a1, a2) + (b1, b2) = (a1 + b1, a2 + b2)
  • subtraction: (a1, a2) - (b1, b2) = (a1 - b1, a2 - b2)
  • scalar multiplication: lambda(a1, a2) = (lambda a1, lambda a2)

From the source example:

  • OA = (-3, 2) = -3e1 + 2e2
  • OB = (4, -1) = 4e1 - e2
  • AB = OB - OA = (7, -3) = 7e1 - 3e2

Key words

  • endpoint: the point where the vector finishes
  • subtraction rule: end minus start
  • scalar: a number that multiplies a vector

Examples

  1. Let P(1, 2) and Q(5, 6). Find PQ.
  2. Compute (2, 3) + (4, -1).
  3. Compute 2(3, -2).

Answers

  1. PQ = (5 - 1, 6 - 2) = (4, 4)
  2. (6, 2)
  3. (6, -4)

3. External Division Point

The source also introduces an external division point. If point S divides PQ externally in the ratio m : n, then the position vector of S is

OS = (-n OP + m OQ) / (m - n)

image-20260423112608524

This formula is similar to the internal division formula, but the signs are different. Be careful here.

From the source example:

  • OP = (-3, 2)
  • OQ = (-5, -1)
  • S divides PQ externally in the ratio 2 : 5

Then

OS = (-5/3, 4)

image-20260423112023497

Key words

  • external division: dividing on the outside of the segment
  • ratio: the comparison m : n
  • formula: a fixed rule used to calculate something

我一直的记忆方式都是对边相乘，不过要能够记得，这个是比例，也就是SQ,PQ分别在总长SP中的占比

根据S在PQ内还是外来决定是内分还是外分

Examples

  1. In one short sentence, what is the difference between internal division and external division?
  2. In the external division formula, which vectors are used?

Answers

  1. Internal division is between the two points, but external division is on an extension line.
  2. OP and OQ

4. Components in Space

In 3D space, a vector has three components. If point A has coordinates (a1, a2, a3), then

a = OA = (a1, a2, a3)

The basic vectors are:

  • e1 = (1, 0, 0)
  • e2 = (0, 1, 0)
  • e3 = (0, 0, 1)

So we can write

a = a1e1 + a2e2 + a3e3

For points P(p1, p2, p3) and Q(q1, q2, q3),

PQ = (q1 - p1, q2 - p2, q3 - p3)

Vector operations in space are also done component by component.

From the source example:

  • P(-2, -3, 5)
  • Q(4, 1, -7)
  • R(3, 6, 2)

Then

  • PQ = (6, 4, -12)
  • PR = (5, 9, -3)
  • 2PQ - 3PR = (-3, -19, -15)

Key words

  • 3D space: space with x, y, and z
  • coordinate: a number showing position
  • component form: writing a vector as a list of coordinates

Examples

  1. Write (2, -1, 4) in basic-vector form.
  2. If P(1, 2, 3) and Q(4, 6, 5), find PQ.
  3. Compute (1, 2, 3) + (2, -1, 4).

Answers

  1. 2e1 - e2 + 4e3
  2. (3, 4, 2)
  3. (3, 1, 7)

5. Centroid and Linear Combination in Space

For a triangle in space, the centroid is the balance point. If the position vectors of the three vertices are OP, OQ, and OR, then the centroid G satisfies

OG = (OP + OQ + OR) / 3

This is a very useful shortcut.

From the source example:

  • P(-2, 4, 1)
  • Q(4, 5, 2)
  • R(4, 0, 3)

Then

OG = (2, 3, 2)

The source also shows how to express a vector as a linear combination:

OS = xOP + yOQ + zOR

For S(4, 23, 5), the result is

OS = 2OP + 3OQ - OR

Key words

  • centroid: the center point of a triangle
  • linear combination: a sum using numbers times vectors
  • coefficient: the number in front of a vector

Examples

  1. What is the centroid formula for a triangle with position vectors OP, OQ, and OR?
  2. If OP = (1, 0, 0), OQ = (0, 1, 0), and OR = (0, 0, 1), find OG.
  3. In OS = 2OP + 3OQ - OR, what is the coefficient of OQ?

Answers

  1. OG = (OP + OQ + OR) / 3
  2. OG = (1/3, 1/3, 1/3)
  3. 3

Quick Review

  • In 2D, a = (a1, a2) = a1e1 + a2e2.
  • In 3D, a = (a1, a2, a3) = a1e1 + a2e2 + a3e3.
  • To find PQ, always do end minus start.
  • Addition, subtraction, and scalar multiplication are done component by component.
  • The centroid of a triangle is the average of the three position vectors.

Missing or unclear source content

  • The side notes and some small callout text in the images are not important for the main formulas, so they were not translated in detail.