Vector Algebra

Scalar (Dot) Product

Understand

The scalar (dot) product of \(\vec a\) and \(\vec b\) is \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\), a number. In components \(\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3\).

\(\vec a\cdot\vec b=0\iff\vec a\perp\vec b\) (non-zero vectors).
\(\vec a\cdot\vec a=|\vec a|^2\); \(\hat i\cdot\hat i=1\), \(\hat i\cdot\hat j=0\).
Angle: \(\cos\theta=\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}\).
Projection of \(\vec a\) on \(\vec b\): \(\frac{\vec a\cdot\vec b}{|\vec b|}\).

Example: \(\vec a=\hat i+2\hat j+2\hat k\), \(\vec b=2\hat i-\hat j+2\hat k\): \(\vec a\cdot\vec b=2-2+4=4\), \(|\vec a|=|\vec b|=3\), so \(\cos\theta=\frac49\).

Key Concepts

a·b = Σ aᵢbᵢ = |a||b|cos θ.
Perpendicular ⇔ a·b = 0.
Projection of a on b = a·b / |b|.

Formula Bank

Dot product and projection

\[\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3,\quad \text{proj}_{\vec b}\vec a=\frac{\vec a\cdot\vec b}{|\vec b|}\]

Key Points

Useful identities

\(|\vec a+\vec b|^2=|\vec a|^2+|\vec b|^2+2\vec a\cdot\vec b\); \((\vec a+\vec b)\cdot(\vec a-\vec b)=|\vec a|^2-|\vec b|^2\).

Common Mistakes

Projection formula

Projection of \(\vec a\) on \(\vec b\) divides by \(|\vec b|\), not \(|\vec a|\).

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Topic Summary

Dot products give angles, perpendicularity and projections.

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