I have this question on a sample midterm for an introductory course on linear Algebra. Even though, I solved it, I am quite unsure if the solution is correct.
Here's what I did-
- To find vector equation of plane, we need a point on the plane and two non-collinear vectors that are parallel to the plane
- Vector equation of the plane is given as-$$\vec{x}=\vec{x}_0+t_1 \vec{v}_1+t_2 \vec{v}_2$$
- here, $$\vec{x}_0=(1,1,3)$$
- So, essentially, I just have to find two more points on the plane which will help me find the other two vectors in the plane equation.
- So, the diagram looks as follows-
6. and hence, I can select any point changing y and z, keeping x the same(since the plane is perpendicular to x-axis)
7. From the random points , I will be able to find $$\vec{v}_1$$ and $$\vec{v}_2$$
8. and I could substitute that into equation to get the answer.
Best Answer
The vector directed along x axis is $$\vec n=(1,0,0)$$ The vector equation of the plane is $$\vec n\cdot(\vec r-\vec r_0)=0\Leftrightarrow 1\cdot(x-1)+0\cdot(y-1)+0\cdot(z-3)=0$$
The equation of the plane perpendicular to x-axis and passing through (1,1,3) is $$x=1$$