What is the vector equation of a line, L, in 3-D? Write this both as a vector with vector labels, r, and also in vector components.
The vector equation of a line L in 3-D can be written as:
r = a + λd
where:
- r is the position vector of any point on the line
- a is the position vector of a known point on the line
- d is the direction vector of the line
- λ is a scalar parameter that can take any real value
In vector components, the equation becomes:
r = <x, y, z> = <x0, y0, z0> + λ<d1, d2, d3>
where:
- <x0, y0, z0> is the known point on the line
- <d1, d2, d3> is the direction vector of the line.
The vector equation of a line, L, in 3-D can be written as:
1. Vector with labels: r(t) = a + t * d
2. Vector with components:
x(t) = x₀ + t * dx
y(t) = y₀ + t * dy
z(t) = z₀ + t * dz
Where:
- r(t) is the position vector of a point on the line L at parameter value t.
- a = (x₀, y₀, z₀) is a known point on the line.
- d = (dx, dy, dz) is the direction vector of the line.
The vector equation of a line, L, in 3-D can be expressed in two forms - one using vector labels and the other using vector components.
1. Vector Label Form:
In vector label form, we use a position vector, r_0, which represents a point on the line, and a direction vector, v, which gives the direction of the line.
The vector equation of the line, L, can be written as:
r = r_0 + t * v
Here, r is the position vector of any point on the line, and t is a scalar parameter that represents different points on the line.
2. Vector Component Form:
In vector component form, we express the vector equation using the components in each direction. We assume that the components of the position vector, r_0, and the direction vector, v, are known.
The vector equation of the line, L, can be written as:
r = <x, y, z> = <x_0 + t * a, y_0 + t * b, z_0 + t * c>
Here, (x, y, z) are the components of any point on the line, (x_0, y_0, z_0) are the components of a point on the line, and (a, b, c) are the components of the direction vector.
To find the vector equation of a line given two points or a point and a direction vector, follow these steps:
1. Determine the position vector, r_0, which represents a point on the line. This can be any point on the line or one of the given points.
2. Calculate the direction vector, v, by subtracting the components of the two given points or taking the direction vector as given.
3. Substitute the values of r_0 and v into the equation r = r_0 + t * v or r = <x_0 + t * a, y_0 + t * b, z_0 + t * c> to obtain the vector equation of the line in the desired form.
Remember, the vector equation of a line provides a compact representation of the line by combining a point on the line and its direction.