Question

A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (1, 2, 0), and (0, 2, 1). By what angle does the tower now deviate from the vertical?

Answer 1

To determine the tower's deviation from the vertical, the angle between the original vertical z-axis and the new normal vector to the plane defined by the given points must be computed.

Step-by-step explanation:

The problem involves calculating the deviation of a tower from the vertical, which is essentially a geometry problem. Given the coordinates (0, 0, 0), (1, 2, 0), and (0, 2, 1), we can infer that the ground has tilted and is no longer a horizontal plane. To find the angle of deviation, we technically want to find the angle between the old vertical (which would be along the z-axis) and the new normal to the plane defined by the three given points.

First, we find the normal vector to the plane using a cross product of the vectors (1, 2, 0) - (0, 0, 0) and (0, 2, 1) - (0, 0, 0), which gives us the vector representing the new 'vertical' due to the tilt. Once we have the normal vector, we then find the angle between this vector and the original z-axis (which would have been 0, 0, 1). The angle can be calculated using the dot product formula, which involves the arccosine function. Finally, we convert the angle from radians to degrees if necessary.

The calculated angle reveals the deviation of the tower from its original vertical position due to the tilting of the ground.

Answer 2

`\theta=cos^(-1)((2)/(3))`

Step-by-step explanation:

We are given that measurements of three points on the ground gave coordinates of (0,0,0),(1,2,0) and (0,2,1)

We have to find the angle by which the tower now deviate from the vertical

We find cross product of <1,2,0> and <0,2,1>

`<1,2,0> * <0,2,1>=\begin{vmatrix}i&j&k\\1&2&0\\0&2&1\end{vmatrix}`

`<1,2,0> * <0,2,1>=2\hat{i}-\hat{j}+2\hat{k}`

Now, we are finding the angle between
`<1,2,0> * <0,2,1>`and vertical vector <0,0,1>

Angle between two vectors formula

`cos\theta=(a.b)/(\mid a\mid\cdot\mid b\mid )`

Now, using this formula

`cos\theta=(2)/(1\cdot 3)=(2)/(3)`

`\theta=cos^(-1)((2)/(3))`

Hence, the tower deviate from the vertical by the angle
`\theta=cos^(-1)((2)/(3))`