Summary Notes: Vectors
Summary Notes: Vectors
SCALAR PRODUCT (DOT PRODUCT)
\({\mathbf{a} \cdot \mathbf{b} = \left| \mathbf{a} \right|\left| \mathbf{b} \right|\cos\theta}\) |
where \({\theta}\) is the angle between \({\mathbf{a}}\) and \({\mathbf{b}}\) when placed at the same initial points (or the same terminal points), \({0^{\circ} \leq \theta \leq 180^{\circ}}\).
Scalar Product in column vectors:
\({\mathbf{a} \cdot \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = a_1 b_1 + a_2 b_2 + a_3 b_3}\)
For example,
\({\begin{pmatrix} -2 \\ 1 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 5 \\ 0 \end{pmatrix} = -4 + 5 + 0 = 1}\)
PROPERTIES OF SCALAR PRODUCT
• \({\mathbf{a} \cdot \mathbf{b}}\) is a real number (scalar), not a vector.
• \({\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}}\) (Commutative law).
• If either \({\mathbf{a} = \mathbf{0}}\) or \({\mathbf{b} = \mathbf{0}}\), then \({\mathbf{a} \cdot \mathbf{b} = 0}\) (real number zero).
• \({\mathbf{a} \cdot \mathbf{a} = \left| \mathbf{a} \right|^2}\), since \({\mathbf{a}}\) is parallel to itself \({\Rightarrow \theta = 0^{\circ} \Rightarrow \cos 0^{\circ} = 1}\).
• \({\mathbf{a} \cdot \mathbf{b} > 0}\) when \({0^{\circ} \leq \theta < 90^{\circ}}\); \({\mathbf{a} \cdot \mathbf{b} < 0}\) when \({90^{\circ} < \theta \leq 180^{\circ}}\).
• For \({\lambda, \mu \in \mathbb{R}}\),
1. \({\lambda(\mathbf{a} \cdot \mathbf{b}) = (\lambda \mathbf{a}) \cdot \mathbf{b} = \mathbf{a} \cdot (\lambda \mathbf{b})}\);
2. \({\lambda \mathbf{a} \cdot \mu \mathbf{b} = (\lambda \mu) \mathbf{a} \cdot \mathbf{b}}\).
• Distributive law
1. \({\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}}\).
2. \({(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}}\).
3. \({(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{c} + \mathbf{d}) = \mathbf{a} \cdot \mathbf{c} + \mathbf{a} \cdot \mathbf{d} + \mathbf{b} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{d}}\),
specially, \({(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} = \left| \mathbf{a} \right|^2 + 2\mathbf{a} \cdot \mathbf{b} + \left| \mathbf{b} \right|^2}\).
Common Mistake: The scalar product of vectors does NOT satisfy Associative law or Cancellation law: \({(\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c} \neq \mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c})}\) \({\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} \not\Rightarrow \mathbf{b} = \mathbf{c}}\) |
APPLICATIONS OF SCALAR PRODUCT
PARALLELISM TEST
\({\mathbf{a} \cdot \mathbf{b} = \pm \left| \mathbf{a} \right| \left| \mathbf{b} \right| \iff \mathbf{a} \parallel \mathbf{b}}\) |
when \({\theta = 0^{\circ}}\) or \({180^{\circ}}\). Specifically,
when \({\theta = 0^{\circ} \Rightarrow \cos 0^{\circ} = 1}\), \({\mathbf{a}}\) and \({\mathbf{b}}\) are parallel and in the same direction;
when \({\theta = 180^{\circ} \Rightarrow \cos 180^{\circ} = -1}\), \({\mathbf{a}}\) and \({\mathbf{b}}\) are parallel and in the opposite direction.
PERPENDICULARITY TEST
\({\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}}\) |
when \({\theta = 90^{\circ} \Rightarrow \cos 90^{\circ} = 0}\), \({\mathbf{a}}\) and \({\mathbf{b}}\) are perpendicular to each other.
ANGLE BETWEEN TWO VECTORS
\({\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|} \Rightarrow \theta = \cos^{-1} \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|} \right)}\) |
LENGTH OF PROJECTION
The length of projection of \({\mathbf{a}}\) onto \({\mathbf{b}}\), \({l}\), is given by \({l = \left| \mathbf{a} \right| \cos \theta = \left| \mathbf{a} \right| \frac{\mathbf{a} \cdot \mathbf{b}}{\left| \mathbf{a} \right| \left| \mathbf{b} \right|}}\), and so
\({l = \frac{\left| \mathbf{a} \cdot \mathbf{b} \right|}{\left| \mathbf{b} \right|} = \left| \mathbf{a} \cdot \hat{\mathbf{b}} \right|}\) |
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