Grade 11 Math Ch-9 Notes

Chapter 9: Introduction to Vectors

Introduction to Vector

A vector is a quantity having both magnitude and direction.

Examples:

• Displacement

• Velocity

• Force

• Acceleration

A scalar is a quantity having only magnitude.

Examples:

• Mass

• Time

• Length

• Temperature

• Speed

Difference between scalar and vector:

Scalar	Vector
Magnitude only	Magnitude and direction
No direction	Has direction
Example: 5 kg	Example: 5 m east

Vector notation:

• Vector: →AB or a

• Magnitude of vector: |a|

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9.1 Geometric Vectors

A geometric vector is represented by a directed line segment.

Characteristics:

1. Magnitude (length)

2. Direction

Types of vectors:

1. Equal vectors

• Same magnitude and same direction

2. Zero vector

• Magnitude = 0

• Direction undefined

3. Unit vector

• Magnitude = 1

If vector a has magnitude |a|,

Unit vector of a:

4. Parallel vectors

• Vectors having the same or opposite direction

5. Opposite vectors

• Same magnitude but opposite directions

Operations on vectors:

Vector Addition

Triangle law:

If vector a and b are joined head-to-tail:

Resultant vector:

a+b

Vector Subtraction

a−b = a + (−b)

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9.2 Applications to Elementary Geometry

Vectors can be used to prove geometrical properties.

Applications:

Midpoint theorem

If M is midpoint of AB:

AM = MB

Parallel line property

If two vectors are proportional:

a=kb

Then vectors are parallel.

Collinear points

Points A, B, C are collinear if:

AB = kBC

where k is a constant.

Triangle property

For triangle ABC:

Applications:

• Proving midpoint theorem

• Proving parallel lines

• Showing points lie on one line

• Solving geometrical problems

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9.3 Position Vectors

The vector drawn from origin O to a point P is called a position vector.

If point P(x,y):

Position vector:

where:

• î = unit vector along x-axis

• ĵ = unit vector along y-axis

Examples:

Point A(3,4)

Position vector:

OA = 3i + 4j

For points A and B:

AB = OB − OA

Properties:

1. Position vectors locate points in a plane.

2. They simplify vector calculations.

3. They help find distance and midpoint.

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9.4 Two-Dimensional Vectors

Two-dimensional vectors lie on the x–y plane.

General form:

Magnitude:

Addition:

Subtraction:

Scalar multiplication:

If k is a scalar:

ka = (kx)i + (ky)j

Example:

a = 2i + 3j
b = 4i + 5j

Then:

a + b

= (2+4)i + (3+5)j

= 6i + 8j

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Important Formulas (Quick Revision)

These notes summarize the chapter for study and exam revision. (My Awady Education)