91影视

In the given exercises use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

The given points are$P(4,3,鈭�2),Q(0,0,3),R(鈭�1,3,6)$

We have$P(4,3,鈭�2),Q(0,0,3),R(鈭�1,3,6)$

Now

role="math" localid="1649673597847" $$\begin{gathered} \stackrel{鈫�}{PQ}=(0-4,0-3,3-(-2\left)\right)=(-4,-3,5) \\ \stackrel{鈫�}{PR}=(-1-4,3-3,6-(-2\left)\right)=(-5,0,8) \end{gathered}$$

$$\begin{gathered} \stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}=\left|\begin{array}{ccc}i& j& k\\ -4& -3& 5\\ -5& 0& 8\end{array}\right| \\ \stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}=i\left|\begin{array}{cc}-3& 5\\ 0& 8\end{array}\right|-j\left|\begin{array}{cc}-4& 5\\ -5& 8\end{array}\right|+k\left|\begin{array}{cc}-4& -3\\ -5& 0\end{array}\right| \\ \stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}=i\left\{\right(-3)脳8-0脳5\}-j\left\{\right(-4)脳8-5脳(-5\left)\right\}+k\left\{\right(-4)脳0-(-3)脳(-5\left)\right\} \\ \stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}=i(-24-0)-j(-32+25)+k(0-15) \\ \stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}=-24i+7j-15k \end{gathered}$$

The points are$(-24,7,-15)$

So,

$$\begin{gathered} ||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}||=\sqrt{{(-24)}^2+{\left(7\right)}^2+{(-15)}^2} \\ ||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}||=\sqrt{576+49+225} \\ ||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}||=卤\sqrt{850} \\ ||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}||=卤5\sqrt{34} \end{gathered}$$

Required vector

$$\begin{gathered} \frac{\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}}{||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}||}=\frac{(-24,7,-15)}{卤5\sqrt{34}} \\ \frac{\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}}{||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}||}=卤\frac{1}{5\sqrt{34}}(-24,7,-15) \end{gathered}$$

$$\begin{gathered} \mathrm{Area}鈭�ABC=\frac{1}{2}||\stackrel{鈫�}{PQ}脳\stackrel{鈫�}{PR}|| \\ \mathrm{Area}鈭�ABC=\frac{1}{2}||5\sqrt{34}|| \\ \mathrm{Area}鈭�ABC=\frac{5\sqrt{34}}{2} \end{gathered}$$