Good morning,
I'll do the 1st example and leave the rest for you:
Equation of the plane: $\displaystyle p: \begin{pmatrix}2\\1\\-1\end{pmatrix} \cdot \vec x = 3$
Equation of the line: $\displaystyle l: \vec r = \begin{pmatrix}1\\6\\-1\end{pmatrix} + \lambda \cdot \begin{pmatrix}-1\\2\\-3\end{pmatrix}$
To determine the points of intersection replace $\displaystyle \vec x$ in p by $\displaystyle \vec r$ of l and solve for $\displaystyle \lambda$:
$\displaystyle \begin{pmatrix}2\\1\\-1\end{pmatrix} \cdot \left( \begin{pmatrix}1\\6\\-1\end{pmatrix} + \lambda \cdot \begin{pmatrix}-1\\2\\-3\end{pmatrix} \right) = 3~\implies~9+3 \lambda = 3~\implies~ \lambda = -2$
Replace $\displaystyle \lambda$ by (-2) in $\displaystyle \vec r$ and you'll get $\displaystyle \begin{pmatrix}3\\2\\5\end{pmatrix} $
To do the following examples you should know which value of $\displaystyle \lambda$ you'll get if
- the line is parallel to the plane (no point of intersection!)
- the line lies in the plane (unlimited number of points of intersection)
In my opinion you haven't done these questions
You are asked to solve equations but you have scribbled down some calculations. To part b):
The final line of the equation should read
$\displaystyle \underbrace{-2}_{\text{your result}} = 3$
So which value has $\displaystyle \lambda$? What does that mean if you are looking for points of intersections?
To part c):
The final line of the equation should read
$\displaystyle \underbrace{3}_{\text{your result}} = 3$
So which value has $\displaystyle \lambda$? What does that mean if you are looking for points of intersections?
so part part c has \lambda . part b has no \lambda. so for the equation has no \lambda means the line is parallel? the line has \lambda=1 means the line has unlimited point of intersection. ? well, i cant understand why \lambda = 1, the equation has unlimited point of intersection?
Good morning,
I guess that you do such excercises in school. Would your teacher accept your explanation as a correct one? Hmmm?
This equation is false. There doesn't exist a value of $\displaystyle \lambda$ which will turn this equation to be true. So if there isn't an appropriate $\displaystyle \lambda$ then there isn't any point of intersection. Therefore the line and the plane must be parallel.The final line of the equation should read
$\displaystyle \underbrace{-2}_{\text{your result}} = 3$
So which value has \lambda? What does that mean if you are looking for points of intersections?
Now you use my argumentation to show that in part c) you can take every real number for $\displaystyle \lambda$. And then you have to deduce what this means geometrically.