Answer
See the explanation
Work Step by Step
To find equations for lines, line segments, and planes in space, we typically use vector and scalar equations.
1. Equations for Lines:
A line in space can be represented by a vector equation or a parametric equation. The vector equation of a line is given by:
r = a + tb,
where r is a position vector on the line, a is a known point on the line, t is a scalar parameter, and b is the direction vector of the line.
Example: Let's say we have a line passing through the point (1, 2, 3) and has a direction vector of (2, -1, 4). The vector equation of this line would be:
r = (1, 2, 3) + t(2, -1, 4).
2. Equations for Line Segments:
A line segment is a portion of a line with two endpoints. To find the equation of a line segment, we can use the parametric equation of a line and restrict the parameter within a specific range.
Example: Consider a line segment with endpoints A(1, 2, 3) and B(4, 5, 6). The parametric equation of this line segment would be:
r = (1, 2, 3) + t((4, 5, 6) - (1, 2, 3)), where 0 ≤ t ≤ 1.
3. Equations for Planes:
A plane in space can be represented by a scalar equation or a vector equation. The scalar equation of a plane is given by:
ax + by + cz = d,
where a, b, and c are the coefficients of the variables x, y, and z, respectively, and d is a constant.
Example: Let's say we have a plane with coefficients a = 2, b = -1, c = 3, and constant d = 4. The scalar equation of this plane would be:
2x - y + 3z = 4.
It is not possible to express a line in space by a single equation because a line extends infinitely in both directions. However, a plane can be expressed by a single equation, as shown in the example above.
