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Vectors (Metric)

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

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 \(z\).

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.

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.

Distance of a plane from the origin, and distance between two parallel planes.

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.