3D Vector question

[Antroph1c]

Hello, I have some problem with solving this question

 

[pka]

Hello, I have some problem with solving this question
View attachment 31315

Can you show that $P:(1,2,-1)$ is on the line $\bf L~?$
Can you explain why $\overrightarrow d=<2,-1,3>\times <1,10,0>$ is the direction vector for $\bf L~?$
Can you explain why the line ${\bf L(t)}= P+t~\overrightarrow d~?$
$$$$

 

[lex]

Perhaps solve the 4 equations given:
(1) 2x - y + 3z = -3
(2) x + 10y = 21
(3) y = 2x
(4) 7x + z = 6
The point you get will therefore lie on both lines.
(Substitute (3) into (1) and (2) to find x, y, and z - then check that (4) works with those values).

Now you know a point on each line (1, 2, -1). You can easily substitute these values to see that this point lies on the plane x + 3y + z = 6
Just pick another point on each line and make sure they lie on the plane too and then you know both lines will lie in the plane.
(To get a point on the line of intersection of 2x - y +3z =-3 and x + 10y = 21, just let e.g. y=0 and work out an x and a z value; all you want is a point that works.
Do the same for the other pair of planes 2x - y = 0, 7x + z = 6)