Chapter 8.4, Problem 37E

a.

To determine

To find: A non-zero vector, representing the monument, perpendicular to the plane defined by given points.

Answer to Problem 37E

The vector $5i+7j+10k$ is perpendicular to the plane defined by the given points.

Explanation of Solution

Given:

The park is determined by a plane which contains the points $\left(1,0,3\right),\left(2,5,0\right)$ and $\left(3,1,4\right)$ .

Calculation:

Let the points contains by a plane is $A\left(1,0,3\right),B\left(2,5,0\right)$ and $C\left(3,1,4\right)$ .

One of the feature of the park is a monument that must be perpendicular to the ground. To represent the monument perpendicular to the ground we have to find the non-zero vector.

The cross product of any two vector in a plane is perpendicular to the plane determined by that plane.

Hence,

  $A\overrightarrow{B}\times A\overrightarrow{C}$ is perpendicular to the plane determined by the points.

Now,

  $\begin{array}{c}A\overrightarrow{B}=O\overrightarrow{B}-O\overrightarrow{A}\\ =\left(2i+5j+0k\right)-\left(i+0j+3k\right)\\ =i+5j-3k\end{array}$

Similarly,

  $\begin{array}{c}A\overrightarrow{C}=O\overrightarrow{C}-O\overrightarrow{A}\\ =\left(3i+j+4k\right)-\left(i+0j+3k\right)\\ =2i+k\end{array}$

  $\begin{array}{c}A\overrightarrow{B}\times A\overrightarrow{C}=\left|\begin{array}{l}ijk\\ 15-3\\ 201\end{array}\right|\\ =i\left(5\times 1-0\times -3\right)-j\left(1\times 1-2\times -3\right)+k\left(1\times 0-5\times 2\right)\\ =5i-7j-10K\end{array}$

Therefore, the vector $5i+7j+10k$ is perpendicular to the plane defined by the given points.

b.

To determine

To find: The criteria so that we know that the vector is perpendicular to the plane defining by the park.

Answer to Problem 37E

The answer is shown below.

Explanation of Solution

Given:

The park is determined by a plane which contains the points $\left(1,0,3\right),\left(2,5,0\right)$ and $\left(3,1,4\right)$ .

Calculation:

If two vectors are perpendicular then the dot product is zero and the cross product of any two vector is always perpendicular to both vectors and perpendicular to the plane in which these two vector lies.

Therefore, the information is given above.