Vectors

• Vectors as displacements in the plane and in three dimensions.
• Components of a vector; column  representation;

Algebraic and geometric approaches to the following:

• the sum and difference of two vectors; the zero vector, the vector −v ;
• multiplication by a scalar, kv ; parallel vectors;
• magnitude of a vector, |v| ;
• unit vectors; base vectors; i, j and k;
• position vectors 
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• Link to three-dimensional geometry, x, y and z- axes.
• Components are with respect to the unit vectors i, j and k (standard basis).
• Applications to simple geometric figures are essential.
• The difference of v and w is v − w = v + (−w) . Vector sums and differences can be represented by the diagonals of a parallelogram.
• Multiplication by a scalar can be illustrated by enlargement.
• Distance between points A and B is the magnitude of 

• Appl: Physics 1.3.2 (vector sums and differences) Physics 2.2.2, 2.2.3 (vector resultants).
• TOK: How do we relate a theory to the author? Who developed vector analysis: JW Gibbs or O Heaviside?

• The scalar product of two vectors.
• Perpendicular vectors; parallel vectors.
• The angle between two vectors.

• The scalar product is also known as the “dot product”.
• Link to 3.6, cosine rule.
• For non-zero vectors, v w = 0 is equivalent to the vectors being perpendicular For parallel vectors, w = kv , |v ⋅w| = |v| |w| .

• Relevance of a (position) and b (direction).
• Interpretation of t as time and b as velocity, with |b| representing speed.

• Aim 8: Vector theory is used for tracking displacement of objects, including for peaceful and harmful purposes.
• TOK: Are algebra and geometry two separate domains of knowledge? (Vector algebra is a good opportunity to discuss how geometrical properties are described and generalized by algebraic methods.)

• Distinguishing between coincident and parallel lines.
• Finding the point of intersection of two lines.
• Determining whether two lines intersect.