Session 1
Lectures 1 - 4
- 12.1 Three-Dimensional Coordinate Systems
- 12.2 Vectors
- 12.3 The Dot Product
- 12.4 The Cross Product
- 12.5 Equations of Lines and Planes
12.1 Three-Dimensional Coordinate Systems
- Defined as \(\mathbb{R}^3 = \{(x,y,z) \mid x,y,z \in \mathbb{R}\}\)
- We represent any point of space by an ordered triple \((a,b,c)\)
Exercises
12.1 Three-Dimensional Coordinate Systems
- What does the equation \(x = 4\) represent in \(\mathbb{R}^2\)?
- What does the same equation represent in \(\mathbb{R}^3\)?
- Describe and/or sketch the surface in \(\mathbb{R}^3\) represented by the equation \(x+z=2\)
12.2 Vectors
We will define a vector \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\)
- Then the magnitude of \(\mathbf{v}\) is given by \(|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\)
- Then the unit vector (direction) of \(\mathbf{v}\) is given by \(\hat{\mathbf{v}} = \mathbf{v} / |\mathbf{v}|\)
- Vector addition is given by \(\mathbf{v} + \mathbf{u} = \langle v_1 + u_1, v_2 + u_2,..., v_n + u_n \rangle\)
- Scalar multiplication by \(c\) is given by \(c\mathbf{v} = \langle cv_1, cv_2,..., cv_n\rangle\)
- The standard basis vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are unit vectors that point in the direction of the positive \(x\)-, \(y\)-, and \(z\)-axes.
- Recall the Parallelogram Law
12.2 Vectors
cont.
Properties of Vectors
- \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\)
- \(\mathbf{a} + \mathbf{0} = \mathbf{a}\)
- \(c(\mathbf{a} + \mathbf{b}) = c\mathbf{a} + c\mathbf{b}\)
- \((cd)\mathbf{a} = c(d\mathbf{a})\)
- \(\mathbf{a} + (\mathbf{b} + \mathbf{c}) = (\mathbf{a} + \mathbf{b}) + \mathbf{c}\)
- \(\mathbf{a} + (-\mathbf{a}) = \mathbf{0}\)
- \((c+d)\mathbf{a} = c\mathbf{a} + d\mathbf{a}\)
- \(1\mathbf{a} = \mathbf{a}\)
Exercises
12.2 Vectors
- Generate an expression that is equivalent to the vector \(\langle 11, 3, 5 \rangle\) as a linear combination of \(\mathbf{v} = \langle 5, 1, 2 \rangle\) and the standard basis vectors (each vector should appear in your expression).
- Find a vector that has the same direction as \(\langle 6,2,-3 \rangle\) but has length 4.
12.3 The Dot Product
If \(\mathbf{a} = \langle a_1, a_2, ..., a_n \rangle\) and \(\mathbf{b} = \langle b_1, b_2, ..., b_n \rangle\), then the dot product of \(\mathbf{a}\) and \(\mathbf{b}\) is the number \(\mathbf{a} \cdot \mathbf{b}\) given by \(\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n\)
- The dot product measures how aligned two vectors are
- If \(\theta\) is the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\), then \(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos{\theta}\)
- Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal \(\iff \mathbf{a} \cdot \mathbf{b} = 0\)
12.3 The Dot Product
cont.
Properties of the Dot Product
- \(\mathbf{a} \cdot \mathbf{a} = {|\mathbf{a}|}^2\)
- \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\)
- \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}\)
- \((c\mathbf{a}) \cdot \mathbf{b} = c(\mathbf{a} \cdot \mathbf{b}) = \mathbf{a} \cdot (c\mathbf{b})\)
- \(\mathbf{0} \cdot \mathbf{a} = 0\)
12.3 The Dot Product
cont.
Projections
- The scalar projection of \(\mathbf{b}\) onto \(\mathbf{a}\) is given by \(\text{comp}_a \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\)
- The vector projection of \(\mathbf{b}\) onto \(\mathbf{a}\) is given by \(\text{proj}_a \mathbf{b} = \left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\right) \frac{\mathbf{a}}{|\mathbf{a}|}\)
Exercises
12.3 The Dot Product
- Let \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\), and let \(\theta_1\) be the angle \(\mathbf{v}\) makes with the \(x\)-axis, \(\theta_2\) be the angle \(\mathbf{v}\) makes with the \(y\)-axis, and \(\theta_3\) be the angle \(\mathbf{v}\) makes with the \(z\)-axis. Evaluate the following expression: \(\cos^2{\theta_1} + \cos^2{\theta_2} + \cos^2{\theta_3}\)
12.4 The Cross Product
If \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\) then the cross product of \(\mathbf{a}\) and \(\mathbf{b}\) is the vector \(\mathbf{a} \times \mathbf{b}\) given by
\[
\mathbf{a} \times \mathbf{b} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3
\end{vmatrix}
\]
- \(\mathbf{a} \times \mathbf{b}\) is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\)
- \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin{\theta}=\) area of the parallelogram determined by \(\mathbf{a}\) and \(\mathbf{b}\)
- Two nonzero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel \(\iff \mathbf{a} \times \mathbf{b} = 0\)
12.4 The Cross Product
cont.
Properties of the Cross Product
- \(\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}\)
- \((c\mathbf{a}) \times \mathbf{b} = c(\mathbf{a} \times \mathbf{b}) = \mathbf{a} \times (c\mathbf{b})\)
- \(\mathbf{a} \times (\mathbf{b}+\mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\)
- \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}\)
- \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\)
Exercises
12.4 The Cross Product
- Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). Let \(\mathbf{a} = 2\mathbf{j} - 4\mathbf{k}\) and \(\mathbf{b} = -\mathbf{i} +3\mathbf{j} + \mathbf{k}\)
12.5 Equations of Lines and Planes
A line through point \(\mathbf{r}_0\) with direction \(\mathbf{v}\) is given by \(\mathbf{r}(t) = t\mathbf{v} + \mathbf{r}_0\)
- Note that \(\mathbf{r}\) is a function of \(t\)
- By eliminating \(t\) we can yield the symmetric equations (not useful in my experience).
A plane through point \(\mathbf{r}_0\) with normal vector \(\mathbf{n}\) is given by \(\mathbf{n} \cdot (\mathbf{r}-\mathbf{r}_0) = 0\)
- Note that \(\mathbf{r}\) is an arbitrary vector
- Expanding the expression we yield the scalar equation of the plane given by \(a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\)
12.5 Equations of Lines and Planes
cont.
Distances
- The distance \(D\) from point \(P_1(x_1,y_1,z_1)\) to the plane \(ax + by + cz + d = 0\) is given by \(D = \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}\)
Exercises
12.5 Equations of Lines and Planes
Suppose \(P_1\) is a plane that contains the points \((0, 3, 0)\), \((-3, 0, 0)\), and \((2, 6, 1)\).
- Find the equation of the plane \(P_1\)
Let \(P_2\) be the plane that contains the point \((0,2,1)\) and the line \(\mathbf{l}(t) = \langle 2t, t, 1+3t \rangle\)
- Find the angle between \(P_1\) and \(P_2\)
Fin
This concludes session 1.