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# Vectors (Metric)

## Class Details

Tom Baker

Learn, or revise, vector quantities, the scalar product and the vector equations of lines and planes.

## Units

### Introduction to Vectors

Vectors, and their magnitude and direction.

The relation $$|a\,{\bf i}+b\,{\bf j}+c\,{\bf k}|=\sqrt{a^2+b^2+c^2}$$.

"Pure direction": the vector of length 1 parallel to a given vector.

### Vector Equation of a Line

The equation of a line in 3D.

Line through a given point in a given direction; line through two points.

For a 3D line, the equations linking $$x$$, $$y$$ and $$z$$.

Lesson

Are these lines parallel, do they meet at a point or do they miss one another entirely?

### The Scalar Product

The scalar product of two vectors, and using it to calculate angles.

The scalar product $${\bf a}\cdot{\bf b}=a\,b\,\cos\theta$$, and using it to calculate $$\theta$$.

Closest distance of approach between a given line and a given point.

Lesson

### Vector Equation of a Plane

The equation of a plane in 3D.

The equation of a plane through a given point, perpendicular to a given vector.

The scalar product $${\bf a}\times{\bf b}=a\,b\,\sin\theta\,{\bf n}$$, and using it to calculate a vector perpendicular to two given vectors.
The triple product $${\bf a}\cdot({\bf b}\times{\bf c})$$ and its applications.