By the end of Week 2: You would learn how to plot equations in 2 variables in 3-space and how to describe and manipulate with vectors. These are just.

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Presentation transcript:

By the end of Week 2: You would learn how to plot equations in 2 variables in 3-space and how to describe and manipulate with vectors. These are just few examples: find an orthogonal projection of a vector onto a given direction vector represent a vector as the sum of two vectors, one being parallel to a given direction vector and the other being orthogonal find a vector perpendicular to two given vectors find the volume of the a parallelepiped with adjacent sided being given vectors

The Dot Product: Properties of the Dot Product:

The Dot Product Formula:

Direction Angles; Direction Cosines direction cosines of v:

Exercises:

Vector and scalar components along standard unit vectors: 2-space Vector components Scalar components 3-space

The orthogonal projection on a unit vector: Vector components are the orthogonal projections on standard unit vectors:

The orthogonal projections on an arbitrary nonzero vector b: is the projection on the unit vector in the direction of v The vector component orthogonal to b:

Solving work problems:

Proofs involving the dot product:

The cross product of two vectors in 3-space *The determinant is a mnemonic device only!

Properties of the cross product

Geometric properties of the cross product Using the right-hand rule to determine the direction of u x v

Exercises:

Exercises:

The scalar triple product Geometric properties of the scalar triple product

Algebraic properties of the scalar triple product

Exercises:

Exercises: