Vectors advanced

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The previous lesson examined vectors in two dimensions, focusing only on vector addition and subtraction.

However, vectors can also be represented in three dimensions and multiplied. In this lesson, we will explore how to manipulate vectors using these concepts.


To start off, we need to learn a new way of representing vectors using unit vectors—vectors with a magnitude of 1 that point along the coordinate axes (î, ĵ, k̂).

Expressing vectors in unit vector form provides a simple, systematic way to perform vector calculations in both two and three dimensions.

Watch the video to learn how unit vectors work and how to use them.

Now that we have covered unit vectors, we can introduce the multiplication of vectors.

Unlike scalar numbers, vectors can be multiplied in two distinct ways:

  • Dot Product (Scalar Product): Multiplies two vectors to yield a scalar value (A · B = |A||B| cos θ). It measures how much one vector acts in the direction of another, commonly used in physics to calculate work.
  • Cross Product (Vector Product): Multiplies two vectors to yield a new vector (A × B = |A||B| sin θ n̂) perpendicular to both original vectors. It is used to calculate rotational quantities like torque and magnetic forces.
The two videos below examine these processes in detail and show examples of where each is applied in physics.

 

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