1.If a theatre holds 1,800 people, and sells ( 1)/(3) of its seats, how many tickets does it sell?
2.Find the vector parametrization for the line segment that begins
at P ( 2, 7, -1) and ends at Q ( 4, 2, 3 ).
The correct answer and explanation is:
Solutions and Explanation
1. Theatre Ticket Sales
The theatre holds 1,800 people. It sells ( \frac{1}{3} ) of its seats.
To find the number of tickets sold:
[
\frac{1}{3} \times 1800 = \frac{1800}{3} = 600
]
So, the theatre sells 600 tickets.
2. Vector Parametrization of a Line Segment
A line segment from point P (2, 7, -1) to point Q (4, 2, 3) can be parametrized using vector equations.
Vector Form:
The general vector equation for a line segment from P to Q is:
[
\mathbf{r}(t) = (1 – t) \mathbf{P} + t \mathbf{Q}, \quad 0 \leq t \leq 1
]
where:
- ( \mathbf{P} = (2,7,-1) ) (starting point)
- ( \mathbf{Q} = (4,2,3) ) (ending point)
- ( t ) is a parameter that varies from 0 to 1.
Component Form:
[
x(t) = (1 – t)(2) + t(4) = 2 + 2t
]
[
y(t) = (1 – t)(7) + t(2) = 7 – 5t
]
[
z(t) = (1 – t)(-1) + t(3) = -1 + 4t
]
Thus, the vector parametrization of the line segment is:
[
\mathbf{r}(t) = (2 + 2t, 7 – 5t, -1 + 4t), \quad 0 \leq t \leq 1
]
Explanation
The concept of vector parametrization is essential in 3D geometry, as it allows us to describe the path of a moving point along a line. A vector equation defines a position along a line as a function of a parameter, typically ( t ), which ranges from 0 to 1 for a line segment.
The equation is derived by linear interpolation between two given points. The idea behind this is that when ( t = 0 ), the function should return the starting point P (2,7,-1), and when ( t = 1 ), it should return the ending point Q (4,2,3). Intermediate values of ( t ) (e.g., ( t = 0.5 )) give points on the line between P and Q.
The formula:
[
\mathbf{r}(t) = (1 – t) \mathbf{P} + t \mathbf{Q}
]
ensures that the position transitions smoothly between the two points. This method works in any dimension (2D, 3D, or higher).
For each coordinate:
- The x-component follows a linear path from 2 to 4, changing by +2 as ( t ) moves from 0 to 1.
- The y-component moves from 7 to 2, decreasing by 5.
- The z-component starts at -1 and increases to 3, changing by 4.
By plugging different values of ( t ), we get various points along the segment, making it useful in physics, animation, and 3D modeling.
Now, let’s generate an image of this line segment in 3D space.
Here’s the 3D visualization of the parametric line segment connecting points P (2,7,-1) and Q (4,2,3). The red dot represents P, the green dot represents Q, and the blue line is the interpolated path between them. Let me know if you need further clarification!
