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Vector Concepts and Notation

CSEC Mathematics: Understanding Vectors

Essential Understanding: A vector is a quantity that has both magnitude (size) and direction. Unlike scalars which only have magnitude, vectors tell us not just "how much" but also "which way." Vectors are fundamental in physics, engineering, and navigation.

Key Skill: Vector Notation
Exam Focus: Column Vectors
Problem Solving: Magnitude & Direction

What is a Vector?

In mathematics, we work with two types of quantities:

5

Scalar Quantity

Definition: A quantity that has only magnitude (size).

Examples:

  • Temperature: 25°C
  • Mass: 50 kg
  • Speed: 60 km/h
  • Time: 3 hours
  • Distance: 100 m
→

Vector Quantity

Definition: A quantity that has both magnitude AND direction.

Examples:

  • Displacement: 5 km North
  • Velocity: 60 km/h East
  • Force: 10 N downward
  • Acceleration: 9.8 m/s² downward

Key Difference

Speed vs Velocity: Speed is a scalar (e.g., "60 km/h"), while velocity is a vector (e.g., "60 km/h heading North"). The direction makes all the difference!

Vector Notation

Vectors can be represented in several ways. In CSEC Mathematics, you need to know all of these:

1. Arrow Notation

A vector from point A to point B is written as \(\overrightarrow{AB}\) with an arrow above the letters.

A B Vector AB

2. Bold Letter Notation

In textbooks, vectors are often written in bold: a, b, v

When handwriting, we use an underline: a, b, v

3. Column Vector Notation (Most Important for CSEC)

A column vector represents movement in the x-direction (horizontal) and y-direction (vertical):

Column Vector Format

\[\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\]

Where x = horizontal component and y = vertical component

Example: Understanding Column Vectors

The vector \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) means:

  • Move 3 units right (positive x)
  • Move 4 units up (positive y)

The vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) means:

  • Move 2 units left (negative x)
  • Move 5 units up (positive y)

Sign Conventions

Direction Sign Example
Right Positive x (+) \(\begin{pmatrix} 5 \\ 0 \end{pmatrix}\)
Left Negative x (-) \(\begin{pmatrix} -5 \\ 0 \end{pmatrix}\)
Up Positive y (+) \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\)
Down Negative y (-) \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\)

Interactive Vector Explorer

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Explore Vectors

Click anywhere on the grid to create a vector from the origin. Observe how the column vector changes!

Current Vector: \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\)

Magnitude: 0 units

Magnitude of a Vector

The magnitude (or length) of a vector is found using the Pythagorean theorem:

Magnitude Formula

\[|\mathbf{a}| = \sqrt{x^2 + y^2}\]

For vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\)

Worked Example 1

Find the magnitude of \(\mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)

1
Apply the formula: \(|\mathbf{v}| = \sqrt{x^2 + y^2}\)
2
Substitute values: \(|\mathbf{v}| = \sqrt{3^2 + 4^2}\)
3
Calculate: \(|\mathbf{v}| = \sqrt{9 + 16} = \sqrt{25} = 5\)

Answer: The magnitude is 5 units

Worked Example 2

Find the magnitude of \(\mathbf{u} = \begin{pmatrix} -5 \\ 12 \end{pmatrix}\)

1
\(|\mathbf{u}| = \sqrt{(-5)^2 + 12^2}\)
2
\(|\mathbf{u}| = \sqrt{25 + 144} = \sqrt{169} = 13\)

Answer: The magnitude is 13 units

Unit Vectors

A unit vector is a vector with magnitude equal to 1. It is used to indicate direction only.

Finding a Unit Vector

\[\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}\]

Divide each component by the magnitude

Worked Example 3

Find the unit vector in the direction of \(\mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)

1
Find magnitude: \(|\mathbf{v}| = \sqrt{3^2 + 4^2} = 5\)
2
Divide each component by 5: \(\hat{\mathbf{v}} = \frac{1}{5}\begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix}\)

Answer: \(\hat{\mathbf{v}} = \begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix}\) or \(\begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}\)

Equal and Parallel Vectors

=

Equal Vectors

Two vectors are equal if they have:

  • Same magnitude
  • Same direction

Note: Position doesn't matter! Equal vectors can start from different points.

∥

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other:

\(\mathbf{b} = k\mathbf{a}\) where k is a scalar

  • k > 0: Same direction
  • k < 0: Opposite direction

Example: Identifying Parallel Vectors

Are \(\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}\) parallel?

Solution: Check if \(\mathbf{b} = k\mathbf{a}\)

\(\begin{pmatrix} 6 \\ 9 \end{pmatrix} = k\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)

From x-component: \(6 = 2k \Rightarrow k = 3\)

From y-component: \(9 = 3k \Rightarrow k = 3\) ✓

Yes, they are parallel (and \(\mathbf{b} = 3\mathbf{a}\))

Inverse (Negative) Vector

The inverse or negative of a vector has the same magnitude but opposite direction:

\[\text{If } \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} \text{, then } -\mathbf{a} = \begin{pmatrix} -x \\ -y \end{pmatrix}\]
a -a

CSEC Practice Questions

Test Your Understanding

1
Which of the following is a vector quantity?
Mass
Temperature
Velocity
Speed
Explanation: Velocity has both magnitude (speed) and direction, making it a vector. Mass, temperature, and speed only have magnitude, so they are scalars.
2
What is the magnitude of the vector \(\begin{pmatrix} 5 \\ -12 \end{pmatrix}\)?
7
17
13
-7
Solution: \(|\mathbf{v}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\)
3
If \(\mathbf{p} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}\), what is \(-\mathbf{p}\)?
\(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\)
\(\begin{pmatrix} -4 \\ -2 \end{pmatrix}\)
\(\begin{pmatrix} -4 \\ 2 \end{pmatrix}\)
\(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\)
Solution: The negative vector reverses both components: \(-\begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} -4 \\ -2 \end{pmatrix}\)
4
Which vector is parallel to \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\)?
\(\begin{pmatrix} 6 \\ 2 \end{pmatrix}\)
\(\begin{pmatrix} 1 \\ 3 \end{pmatrix}\)
\(\begin{pmatrix} 4 \\ 8 \end{pmatrix}\)
\(\begin{pmatrix} 2 \\ -6 \end{pmatrix}\)
Solution: \(\begin{pmatrix} 1 \\ 3 \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 2 \\ 6 \end{pmatrix}\). Since one is a scalar multiple of the other, they are parallel.
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CSEC Examination Tips

  • Always show your working when finding magnitude - write out the full formula with substitutions.
  • Remember: Magnitude is always positive! Even if the vector components are negative.
  • Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17. Recognizing these can save time!
  • For parallel vectors: Check if the ratio of x-components equals the ratio of y-components.
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