For the vectors \(\uu = \la 3, -1, 4\ra, \vv = \la -2, -2, 5\ra, \ww = \la 0,4,-3\ra \text{,}\) find:

1. \(\displaystyle \vec{u}\dotp(2\vec{v}-\vec{w})\)
2. \(\displaystyle (2\vec{u})\dotp\vec{v}-\vec{u}\dotp\vec{w}\)
3. A value of \(c\) such that \(\uu\) is orthogonal to \(\vv+c\ww\)

Recall: for vectors \(\vv,\ww\text{,}\) \(\vv\dotp\ww = \norm{\vv}\,\norm{\ww}\cos\theta\text{,}\) where \(\theta\) is the angle between \(\vv\) and \(\ww\text{.}\)

Most useful for us: when \(\theta = \pi/2\text{.}\)

• Defined for vectors in \(\R^3\) only.

• Produces a vector rather than a scalar.

• Cross product \(\vv\times \ww\) is orthogonal to both \(\vv\) and \(\ww\text{.}\)

• Definition: if \(\vv = \la v_1,v_2,v_3\ra, \ww = \la w_1,w_2,w_3\ra\text{,}\)

  \begin{equation*} \vv\times\ww = \la v_2w_3-v_3w_2,v_3w_1-v_1w_3,v_1w_2-v_2w_1\ra\text{.} \end{equation*}
  (Easier to remember using “determinant trick”.)

Let \(\uu = \la 3, -1, 2\ra, \vv = \la 0, 2, -1\ra, \ww = \la -2, 0, 4\ra\text{.}\) Find:

• \(\uu\times \vv\)

• \(\vv\times \ww\)

• \(\ww\times \vv\)

1. \(\vv\times \ww = -\ww\times \vv\)

2. \(\vv\times (c\ww) = (c\vv)\times \ww = c(\vv\times \ww)\)

3. \(\uu\times (\vv+\ww)=\uu\times \vv+\uu\times \ww\)

4. \(\vv\times \ww\) is orthogonal to both \(\vv\) and \(\ww\text{.}\)

If the angle between \(\vv\) and \(\ww\) is \(\theta\text{,}\)

Useful to note: if \(\vv\) and \(\ww\) form 2 of 4 sides of a parallelogram, that parallelogram has area \(A = \norm{\vv}\,\norm{\ww}\sin\theta\text{.}\)

Let \(P=(0,2,-1), Q = (3, 1, -2), R = (4, -2, 0), S = (7, -3, -1)\text{.}\)

Verify that the quadrilateral with these vertices is a parallelogram, and find its area.

What about the triangle \(\Delta PQR\text{?}\)

We all know lines in the plane: \(y=mx+b\) gives a line with slope \(m\) passing through \((0,b)\text{.}\) Does slope make sense in \(\R^3\text{?}\)

• In \(\R^3\text{,}\) to specify a line we need a point (on the line), and a direction (vector).

• Suppose \(P_0 = (x_0,y_0,z_0)\) and \(P=(x,y,z)\) are two points on a line in the direction of a vector \(\vv = \la a,b,c\ra\text{.}\) What can we say about the vector \(\overrightarrow{P_0P}\text{?}\)

Points on a line in space are given in terms of a parameter (usually \(t\) — we can think of motion in a straight line, with \(t\) as time).

• The vector equation of a line through \(P_0=(x_0,y_0,z_0)\) in the direction of \(\vv = \la a,b,c\ra\) is

  \begin{equation*} \la x,y,z,\ra = \la x_0,y_0,z_0\ra + t \la a,b,c\ra\text{,} \end{equation*}
  or \(\vec{x}=\vec{x}_0+t\vv\text{,}\) for short.

• Sometimes see \(\vec{r}\) or \(\vec{r}(t)\) instead of \(\vec{x}\text{.}\)

• Equating coefficients in the vector equation gives the parametric equations:

  \begin{align*} x \amp = x_0+at\\ y \amp = y_0+bt\\ z \amp = z_0+ct\text{.} \end{align*}

Find the vector equations of the lines:

1. Through the point \(P_0 = (2,-5,1)\) in the direction of \(\vv = \la 6,-7,3\ra\)

2. Through the points \(P=(4,5,-3)\) and \(Q=(-1,8,2)\text{.}\)

3. Through the point \(P_0=(4,5,-9)\) and parallel to the line

   \begin{equation*} \vec{r}(t) = \la 3-4t, -6+8t, 2-7t\ra\text{.} \end{equation*}

• Two lines in \(\R^3\) are parallel if their direction vectors are parallel.

• Not all paralell lines intersect: some are skew.

• Checking for intersection leads to a system of equations. (Be sure to use a different parameter for each line.)

Given vectors \(\uu\) (not equal to \(\vec{0}\)) and \(\vv\text{,}\) often useful to write \(\vv\) as the sum of a vector parallel to \(\uu\text{,}\) and a vector orthogonal to \(\uu\text{.}\)

Vector \(\vec{a}\) called the projection of \(\vv\) onto \(\uu\text{.}\) Notation and formula: