Coordinates Of A Point In Space

To determine the octant in which the point (2, -4, -7) lies, we need to consider the signs of its coordinates.
In an octant, the x, y, and z coordinates can have either positive (+) or negative (-) values.

Given the point (2, -4, -7), we can determine the octant as follows:
First, consider the x-coordinate, which is 2. Since it is positive, we eliminate the octants with negative x-values (4^th, 5^th, 6^th, and 7^th).
Next, consider the y-coordinate, which is -4. Since it is negative, we eliminate the octants with positive y-values (1^st, 2^nd, 5^th, and 6^th).
Lastly, consider the z-coordinate, which is -7. Since it is negative, we eliminate the octants with positive z-values (1^st, 2^nd, 3^rd, and 4^th).

Based on the elimination process, we find that the point (2, -4, -7) lies in the Eighth octant.
Therefore, the octant in which the point (2, -4, -7) lies is the Eighth octant (option A).

To determine the value of β, we first need to find the slope of the line passing through the points (7, 17) and (15, β).
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:

For the given points (7, 17) and (15, β), the slope is:

Now, we know that a line is perpendicular to another line if and only if the product of their slopes is -1.
Therefore, we need to find the slope of the line and determine the value of β that makes the product of the slopes -1.

The given line can be rewritten in slope-intercept form as:

Comparing this equation with the standard slope-intercept form , we can see that the slope of this line is
Now, we have:
slope of the line passing through (7, 17) and
slope of the line

To find β, we set the product of the slopes equal to -1:

Simplifying the equation:

Multiplying both sides by
(β - 17) = -8 * (3/2)

Solving for β:

Therefore, β equals 5 (option C).