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Section 3 Key Concepts

The angle between two vectors.

With their tails are placed at the same point, the angle between two non-zero, non-parallel vectors \(\vec{w}\) and \(\vec{v}\) is the angle \(\theta\) with \(0\leq \theta \leq \pi\) between \(\vec{w}\) and \(\vec{v}\) in the plane that contains them. In the spacial case when \(\vec{v}=\vec{0}\) or \(\vec{w}=\vec{0}\text{,}\) we say that \(\theta =\pi /2\text{.}\)

  • \(\vec{w}\) and \(\vec{v}\) are parallel if the angle between them is \(0\) or \(\pi\text{.}\)
  • \(\vec{w}\) and \(\vec{v}\) are perpendicular or orthogonal if the angle between them if \(\pi /2\) (so, in particular the zero vector is orthogonal to all vectors).
The Definition of the Dot Product.

  • The Definition of the Dot Product: Suppose \(\vec{w}=(u_{1},\dots ,n_{n})\) and \(\vec{v}=(v_{1},\dots ,v_{n})\) are two vectors in \({\mathbb R}^n\text{.}\) The dot product \(\vec{w} \cdot \vec{v}\) is defined as:

    • Algebraic Definition: \(\vec{w} \cdot \vec{v} =u_{1}v_{1}+u_{2}v_{2}+\dots +u_{n}v_{n}\text{.}\)
    • Geometric Definition: \(\vec{w} \cdot \vec{v} =||\vec{w} ||||\vec{v} ||\cos{\theta}\) where \(\theta\) is the angle between them.
  • Geometric Interpretation of the Dot Product. The dot product of two vectors is a numerical measure of how their directions are related. Specifically, it is large and positive (relative to their magnitudes) if they point in roughly the same direction, large and negative if roughly opposite directions, and close to \(0\) if they are roughly perpendicular.
Properties and Applications of the Dot Product.

  • If \(\vec{w}\text{,}\) \(\vec{v}\text{,}\) and \(\vec{u}\) are vectors and \(c\) is a scalar, then the following properties are easy to verify using the algebraic definition of the dot product:

    • \(\vec{v} \cdot \vec{v} =||\vec{v}||^2\)
    • \(\vec{w} \cdot \vec{v} =\vec{v} \cdot \vec{w}\)
    • \(\vec{w} \cdot (\vec{v} +\vec{u}) =\vec{w} \cdot \vec{v} +\vec{w} \cdot \vec{u}\)
    • \(c(\vec{w} \cdot \vec{v}) =(c\vec{w}) \cdot \vec{v}=\vec{w} \cdot (c\vec{v})\)
  • Finding Angles Between Vectors: The angle \(\theta \) between nonzero vectors \(\vec{w}\) and \(\vec{v}\) is \(\theta =\arccos{\left( \frac{\vec{w} \cdot \vec{v}}{||\vec{w}|| ||\vec{v}||}\right)}.\)
  • Test for Orthogonality: Two vectors \(\vec{w}\) and \(\vec{v}\) are orthogonal if and only if \(\vec{w} \cdot \vec{v} =0\text{.}\)
  • Finding a Scalar Equation for a Plane: An equation of the plane with normal vector \(\vec{n} =a\vec{i} +b\vec{j} +c\vec{k}\) and containing the point \(P_0=(x_{0},y_{0},z_{0})\) is \(a(x-x_{0}) +b(y-y_{0}) +c(x-z_{0})=0.\) For \(d=ax_0+by_0+cz_0\text{,}\) this equation can be rewritten as \(ax +by +cx=d.\)
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