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Vector Mathematics in 3D
An Introduction to 3D Vectors
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Prof. David Bernstein
James Madison University
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Computer Science Department
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bernstdh@jmu.edu
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Some Notation
- The Set of Real Numbers:
- \mathbb{R}^{1} \doteq \mathbb{R}
- The Set of Ordered Triples of Real Numbers:
- \mathbb{R}^{3} \doteq \mathbb{R} \times \mathbb{R} \times \mathbb{R}
Members of \mathbb{R}^{3}
- The Notation You Are Most Familiar With:
- A More Convenient Notation:
- \bs{p} = (p_{x}, p_{y}, p_{z})
- A Still More Convenient Notation:
- \bs{p} = (p_{1}, p_{2}, p_{3})
Two Things to be Aware Of
- Orientation:
- It sometimes matters whether we think of the ordered pair as
a "row" or a "column", though we won't distinguish for now
- An example of a "row": \bs{p} = [p_1 \quad p_2 \quad p_3]
- An example of a "column": \bs{p} = \left[ \begin{array}{c}p_1 \\ p_2 \\ p_3 \end{array} \right]
- Points and Directions:
- In is sometimes important to distinguish between
different kinds of ordered pairs (e.g., points and directions)
- We won't distinguish for now except in visualizations
Visualization (cont.)
Directions
Vector Fields
- Definition:
- A vector field is a map
F: A \subset \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}
that assigns to each point \bs{p} \in A
a vector F(\bs{p})
- Visualization:
Multiplication by a Scalar
- Definition:
- Given \alpha \in \mathbb{R} and
\bs{p} \in \mathbb{R}^{3}
- \alpha \bs{p} = (\alpha p_{1}, \alpha p_{2}, \alpha p_{3})
- Visualization:
Addition
- Definition:
- Given \bs{q} \in \mathbb{R}^{3} and \bs{r} \in \mathbb{R}^{3}
- \bs{q} + \bs{r} = (q_{1}+r_{1}, q_{2}+r_{2}, q_{3}+r_{3})
- Visualization:
Subtraction
- Definition:
- Given \bs{q} \in \mathbb{R}^{3} and \bs{r} \in \mathbb{R}^{3}
- \bs{q} - \bs{r} = (q_{1}-r_{1}, q_{2}-r_{2}, q_{3}-r_{3})
- Visualization:
The Standard Basis
- Three Important Vectors:
- \bs{i} = (1, 0, 0)
- \bs{j} = (0, 1, 0)
- \bs{k} = (0, 0, 1)
- A General Representation of Vectors in \mathbb{R}^3:
- Suppose \bs{q} = (q_{1}, q_{2}, q_{3}), then
- \bs{q} = q_{1} \bs{i} + q_{2} \bs{j} + q_{3} \bs{k}
The Standard Basis (cont.)
Visualization
The Standard Basis (cont.)
- An Important Observation:
- Every vector \bs{p} can be written as
\bs{p} = \alpha \bs{i} + \beta \bs{j} + \gamma \bs{k}
when \alpha, \beta
and \gamma are chosen appropriately
- As in Two Dimensions:
- \bs{p} is then said to be a
linear combination
of \bs{i}, \bs{j}, and \bs{k}
- \bs{i}, \bs{j}, and \bs{k}
are said to be
linearly independent
The Inner (Dot) Product
- Two Notations:
- \bs{p} \cdot \bs{q}
- \lt \bs{p}, \bs{q} \gt
- Definition:
- Given \bs{p} \in \mathbb{R}^{3} and \bs{q} \in \mathbb{R}^{3}
- \bs{p} \cdot \bs{q} = \sum_{i=1}^{3} p_{i} q_{i}
= p_{1} q_{1} + p_{2} q_{2} + p_{3} q_{3}
The Euclidean Norm (Length) of a Vector
- The (Euclidean) Length of a Vector:
- ||\bs{p}|| = \sqrt{\sum_{i=1}^{3} p_{i}^{2}} =
\sqrt{p_{1}^{2} + p_{2}^2 + p_{2}^2}
- An Observation:
- \bs{p} \cdot \bs{p} = p_{1}^{2} + p_{2}^2 + p_{3}^2
- Substituting:
- ||\bs{p}|| = (\bs{p} \cdot \bs{p})^{\frac{1}{2}}
Unit Vector
- Definition:
- A vector with a norm (i.e., length) of 1
- Three Examples Revisited:
- \bs{i} = (1, 0, 0)
- \bs{j} = (0, 1, 0)
- \bs{k} = (0, 0, 1)
Normalization
- Definition:
- The normalization of a vector \bs{p}
is \frac{\bs{p}}{||\bs{p}||}
(i.e., the vector divided by its norm)
- An Observation:
- ||\bs{p}|| is a scalar so normalization involves
multiplication by a scalar
- Relationship to Unit Vectors:
-
\left\| \frac{\bs{p}}{||\bs{p}||} \right\| = 1
The Angle Formed by Vectors
- Getting Started:
- The Plane Formed by the Vectors:
- \bs{p} and \bs{q} define a plane
- So, what we learned about 2D vectors applies here
- From the Law of Cosines:
- \theta = \cos^{-1} \frac{\bs{p} \cdot \bs{q}}{||\bs{p}|| \enspace||\bs{q}||}
Cross Product
- Definition:
- Given two vectors \bs{v}, \bs{w} \in \mathbb{R}^{3},
the vector
\bs{u} \in \mathbb{R}^{3} defined as:
- \bs{u} = (v_{2}w_{3} - w_{2}v_{3},
v_{3}w_{1} - w_{3}v_{1},
v_{1}w_{2} - w_{1}v_{2})
- is called the cross product of
\bs{v} and \bs{w}
- An Example:
- Let \bs{v} = \left[ \begin{array}{c}1 \\ 0 \\ 2\end{array}\right]
and \bs{w} = \left[ \begin{array}{c}5 \\ 3 \\ 4\end{array}\right]. Then:
- \bs{u} = \bs{v} \times \bs{w} = \left[ \begin{array}{c}0 \cdot 4 - 3 \cdot 2 \\
2 \cdot 5 - 4 \cdot 1 \\
1 \cdot 3 - 5 \cdot 0\end{array}\right]
= \left[ \begin{array}{c}-6 \\ 6 \\ 3\end{array}\right]
- Be Careful:
- Unlike the inner (or dot) product, the cross product
is a vector
- Don't confuse this with the cross product of two sets
- Don't confuse the different uses of the
symbol \times
Cross Product (cont.)
- The Visualization Revisited:
- An Interpretation:
- || \bs{v} \times \bs{w} || = ||\bs{v}|| \enspace ||\bs{w}|| \sin \theta
which is the (unsigned) area of the parallelogram formed
by \bs{v}, \bs{w}, and (\bs{v}+\bs{w})
Cross Products (cont.)
- An Important Property:
- \bs{v} \times \bs{w} is orthogonal to both
\bs{v} and \bs{w} (and, hence, all vectors in
the plane they form)
- Proof:
- (\bs{v} \times \bs{w}) \cdot \bs{v} =
(v_1 v_2 w_3 - v_1 w_2 v_3) + (w_1 v_2 v_3 - v_1 v_2 w_3) +
(v_1 w_2 v_3 - w_1 v_2 v_3) = 0
- (\bs{v} \times \bs{w}) \cdot \bs{w} =
(w_1 v_2 w_3 - w_1 w_2 v_3) + (w_1 w_2 v_3 - v_1 w_2 w_3) +
(v_1 w_2 w_3 - w_1 v_2 w_3) = 0
- An Observation:
- This property is sometimes used as the defition of the cross
product
The Cross Product Revisited
- A Special Case:
- Replace \bs{u} with the unit normal n
- Visualization:
- Interpretation:
- Given the definition of \bs{n} we know that
||\bs{n}|| = 1
- So, ignoring units, the volume of this parallelepiped
is the same as the area of the parallelogram
- We can use ||\bs{n}|| \cos \alpha ||\bs{v} \times \bs{w}|| =
\bs{n} \cdot (\bs{v} \times \bs{w})
as the signed area of the
parallelogram
There's Always More to Learn
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