inknit said:For example.
L1: x = 11+3t y = 7+t z = 9+2t
L2: x=-6+4t y=-2+3t z=-7+5t
I was given this problem, and technically these lines don't intersect, right?
inknit said:Alright so, they don't intersect correct? B/c if you replace the variable in L1 with let's say "s" they intersect at some point.
They don't happen to intersect (and not just "technically") but not because the use the same parameter. A parameter has no meaning outside the equation itself. So as not to confuse yourself, it would be better to change one of them to, say, "s". To determine if they intersect, try to solve x= 11+ 3t= -6+ 4s, y= 7+ t= -2+ 3s, z= 9+ 2t= -7+ 5s for s and t.inknit said:For example.
L1: x = 11+3t y = 7+t z = 9+2t
L2: x=-6+4t y=-2+3t z=-7+5t
I was given this problem, and technically these lines don't intersect, right?
Yes, any line in 3D space can be parametrized using a single variable, such as t. This allows for the coordinates of any point on the line to be expressed as a function of t, making it easier to manipulate and analyze mathematically.
To determine the parametric equations for a line in 3D space, you can use a point on the line (P) and the direction vector (v) of the line. The parametric equations are then x = Px + tvx, y = Py + tvy, and z = Pz + tvz, where P is the point and v is the direction vector.
Yes, a line in 3D space can have multiple parametrizations. This is because there are infinite ways to represent the same line using different values for the parameter. However, the simplest and most common parametrization is using a single variable, such as t.
No, not all lines in 3D space have unique parametrizations. Some lines, such as parallel lines, may have the same parametric equations but different values for the parameter. However, each point on the line can still be uniquely described using the parametric equations.
The parametric equations for lines in 3D space have an additional variable (z) compared to those in 2D space. This is because lines in 3D space have three dimensions, while lines in 2D space only have two dimensions. Additionally, the direction vector in 3D space has three components instead of two.