TensorSpeed

1. Dot Product – Deeper Look

The dot product of two vectors is a single number (scalar) that measures how much they point in the same direction.

Algebraic formula (2D): \(\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y\)

Geometric formula: \(\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos \theta\)

Key facts:

If dot product > 0 → acute angle (point same-ish direction)
If dot product = 0 → perpendicular (90°)
If dot product < 0 → obtuse angle (opposite-ish direction)

Exercise 1

2. Geometric Meaning & Angle

From the geometric formula, we can find the angle between two vectors:

\[ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|} \]

Example: If dot product = 0, cos θ = 0 → θ = 90° (perpendicular).

Exercise 2

Which pairs of vectors are perpendicular? (Select all that apply)

\(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\) \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\) \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) and \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) \(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\)

3. Projection of One Vector onto Another

The scalar projection of \(\vec{a}\) onto \(\vec{b}\) is:

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b} \]

It tells how much of \(\vec{a}\) lies in the direction of \(\vec{b}\).

Exercise 3

What is the scalar projection of \(\vec{a} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}\) onto \(\vec{b} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}\)?