Equation of the Straight Line from Origin to Point 10-1

When dealing with the equation of a line in three-dimensional space, it is crucial to understand the parameters involved. In this case, we are interested in the line passing through the origin (0,0,0) and the point (10, -1, 0).

Direction Ratios of the Line

The direction ratios of the line can be derived from the coordinates of the given points. For the line passing through the origin (0,0,0) and the point (10,-1,0), the direction ratios are:

1, 0, -1

This means that the line can be described by the ratios of the differences in the coordinates of the points. Here, the difference in the x-coordinate is 10 - 0 10, the difference in the y-coordinate is -1 - 0 -1, and the difference in the z-coordinate is 0 - 0 0. Thus, the direction ratios are 10, -1, 0.

Equation of the Straight Line

The equation of the line can be derived using the direction ratios and the given point. The general form of the equation for a line in three-dimensional space is:

frac{x - x1}{a} frac{y - y1}{b} frac{z - z1}{c}

where (a, b, c) are the direction ratios and (x1, y1, z1) is any point on the line. In this case, the line passes through the origin (0, 0, 0) and the point (10, -1, 0), so we use 0 as x1, y1, and z1, and the direction ratios 10, -1, and 0.

frac{x - 0}{10} frac{y - 0}{-1} frac{z - 0}{0}

However, since division by zero is undefined, we need to consider the behavior of the equation. In this case, the z-coordinate is always zero, which simplifies the equation to:

x/10 y/(-1) z/0, but z is actually 0.

This leads to the simplified equations:

x/10 y/(-1) and z 0.

These represent the line in the xy-plane, where z is always 0.

Alternative Representations

There are also alternative ways to represent this line. One such way is to use the parametric form:

x 10t, y -t, z 0

where t is a parameter. Another representation is:

x/1 y/0 z/-1

This form is valid as long as y is always 0 and x and z follow the direction ratios.

Verification with the Distance Formula

To verify the correctness of the line equation, we can use the distance formula. The distance between two points in 3D space is given by:

d sqrt[(x2 - x1)^2 (y2 - y1)^2 (z2 - z1)^2]

For the line passing through (0,0,0) and (10,-1,0), the distance is:

d sqrt[(10 - 0)^2 (-1 - 0)^2 (0 - 0)^2] sqrt[100 1 0] sqrt(101)

This confirms that the point (10, -1, 0) is indeed 101 units away from the origin, which is consistent with the direction ratios.

Conclusion

In conclusion, the equation of the straight line from the origin to the point (10, -1, 0) can be represented in several forms, including the direction ratios, the simplified equation, and the parametric form. The key is to understand the direction ratios and how they relate to the coordinates of the points on the line.

Keywords: line equation, direction ratios, origin, point 10-1