*Here you’ll find all of our articles with teaching strategies to bring a discussion of borderlands **history into your classroom. We’ll keep this section updated as we add new content.*

**Teaching Borderlands History to Undergraduates, Part I: Framing Your Course**

**Teaching Borderlands History to Undergraduates, Part II: What’s Worked and What Hasn’t**

**Dare to Compare: Attempting Comparative Transnational and Borderlands History**

**Teaching North American Borderlands History Online**

**On Thinking about Borderlands in World History**

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EDUCATION AND DOCTORAL DIPLOMA

Author Mircea Orasanu

These recommendations, which were received at the 2000 EC meeting held just prior to ICME-9, were positive so that the decision made by the ICMI EC to establish two ICMI sponsored awards was officially announced at the ICME-9 General Assembly.

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PEDAGOGY ON LEARNING

ABSTRACT The basis vectors have an important property of linear independent which is defined as follow:

Definition The set of vector is said to be linear independent if and only if the vectors equation has only solution

Definition The set of vector is said to be linear dependent if and only if the vectors equation has non-trivial solution.

(i.e. there exists some such that )

Example Determine whether are linear independent or dependent.

Solution

Example Let and Prove that

(a) and are linearly independent.

(b) any vector in can be expressed as a linear combination of and .

Solution

Example If vectors and are linearly independent, show that and are also linearly independent.

Solution

Example L

1 INTRODUCTION

Definition The scalar product or dot product or inner product of two vectors and , denoted by , is defined as

where is the angle between and .

Remarks By definition of dot product, we can find by .

Example If and angle between and is , then

Theorem Properties of Scalar Product

Let be three vectors and be a scalar. Then we have

(1)

(2)

(3)

(4)

(5) if and if

Theorem If and . Then

(1)

(2) =

=

(3) if and only if .

(4) if and only if .

Example Find the angle between the two vectors and

Solution

Remarks Two non-zero vectors are said to be orthogonal if their scalar product is zero. Obviously, two perpendicular vectors must be orthogonal since , , and so their scalar product is zero. For example, as and are mutually perpendicular, we have

.

Also, as and are unit vectors, .

Example State whether the two vectors and are orthogonal.

Solution

Example Given two points and

and two vectors

and

If is perpendicular to both and , find the values of and .

Solution

Definition If and are vectors in , then the vector product and cross product is the vector defined by

=

=

Example Find , and if and .

Solution

Example Let and Find

(a) (b) (c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

Solution

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EDUCATION AND FORMATION

ABSTRACT

, where , are the two angular coordinates in the standard spherical polar coordinate system.Solve this ODE by reduction of order:

1 INTRODUCTION

OR (a much faster solution!)

Adjacent angles Two coplanar angles that share a vertex and a side but do not overlap

Alternate interior angles

Two angles that lie on opposite sides of a transversal between two lines that the transversal intersects

Altitude of a triangle A perpendicular segment from a vertex of a triangle to the line that contains the opposite side

Angle Two non-collinear rays having the same vertex

Angle of depression

When a point is viewed from a higher point, the angle that the person’s line of sight makes with the horizontal

Angle of elevation

When a point is viewed from a lower point, the angle that the person’s line of sight makes with the horizontal

Example Let and be three coplanar vectors. If and are orthogonal, show that

Solution

Central Force La