Scalar/Dot Product (2)

Video lesson · from 38:381 worked example

One lesson video covers all of Vectors, so it opens at 38:38 for this topic — not from the beginning.

Theory

The dot product (scalar product), denoted $\vec{a} \cdot \vec{b}$ , can be calculated as follows (Polar perspective):

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta

where $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$ . The vectors must both be pointing away from the vertex.

⚠️ Common Examiner Traps

Use the magnitudes, not the components: the geometric form is $|\vec{a}||\vec{b}|\cos\theta$ . Mixing the two forms in one line is a frequent error.
The angle is between the vectors: both must point away from the common vertex. If one points towards it, reverse it first or the angle will be wrong.
Keep magnitudes exact: rounding surds early makes the final angle inaccurate.
Choose the form that suits the question: components when you know them, magnitudes and angle when you are given those.

Worked examples

Example 1

Vectors $\vec{a}$ and $\vec{b}$ have magnitudes 7 and 3 units respectively and are at an angle of $60^\circ$ to each other. What is the value of $\vec{a} \cdot \vec{b}$ ?

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = (7)(3) \cos 60^\circ = 21 \left(\frac{1}{2}\right) = 10.5