That is the difference between a set of three parametric equations for three variables (x, y, and z) with a single parameter (t), and a single linear equation in three variables (x, y, and z). In both cases, the values of x, y, and z can be envisioned as points in 3-dimensional Cartesian space. However, the single linear equation in three variables defines a plane in 3-dimensional space and the three parametric equations define a line in 3-dimensional space. The reason for the difference is that in the single linear equation, one variable (for example z) varies in a way that is dependent on the other two variables (x and y) so for every combination of x and y, there is a corresponding value for z, which would define a height above the point (x,y) in the x-y plane (See "Algebra 18 - Multi-variable Functions"). On the other hand, the three parametric equations each define a single value for x, y, or z as the parameter t varies, so the result is a line in 3 -dimensional space.
so u said that 3 variable equation form a plan because it three dimensional object lets say, but in parametric equation we take only the point on this plane which form a line. if this true the next question will be so why we take that specific line why we dont take any line from that intersect plane (is that because this intersect plane only intersect on this line ?! i think so )
these explenations are priceless, so glad i found this channel
Keep up the good work 👍 I'm always looking for the next episode.
Thank you so much!
I was waiting for the next episode so bad.
Can’t ask for a better explanation
Thank you, i miss you professor
this is related to vector equations in calculus 3!
great explanation!!
Thank you Professor!
i dont understand this "linear equation with three variable describe plane not lines "
That is the difference between a set of three parametric equations for three variables (x, y, and z) with a single parameter (t), and a single linear equation in three variables (x, y, and z). In both cases, the values of x, y, and z can be envisioned as points in 3-dimensional Cartesian space. However, the single linear equation in three variables defines a plane in 3-dimensional space and the three parametric equations define a line in 3-dimensional space. The reason for the difference is that in the single linear equation, one variable (for example z) varies in a way that is dependent on the other two variables (x and y) so for every combination of x and y, there is a corresponding value for z, which would define a height above the point (x,y) in the x-y plane (See "Algebra 18 - Multi-variable Functions"). On the other hand, the three parametric equations each define a single value for x, y, or z as the parameter t varies, so the result is a line in 3 -dimensional space.
so u said that 3 variable equation form a plan because it three dimensional object lets say, but in parametric equation we take only the point on this plane which form a line. if this true the next question will be so why we take that specific line why we dont take any line from that intersect plane (is that because this intersect plane only intersect on this line ?! i think so )
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