Find point where line intersects plane

[jdj333]

Homework Statement

Find the point as which the line intersects the given plane.

Homework Equations

Line: x = y - 1 = 2z
Plane: 4x - y + 3z = 8

The Attempt at a Solution

I understand how to use the cross product, dot product, and find intercepts. This problem is in section 13.5 #45 of the James Stewart Calculus book. I understand the idea but need some help in solving the problem. Thanks!

 

[Дьявол]

line:

$$\frac{x-x_1}{a_1}=\frac{y-y_1}{a_2}=\frac{z-z_1}{a_3}$$

plane:

$$Ax+By+Cz+D=0$$

line:

$$x=x_1+ta_1 ; y=y_1+ta_2 ; z=z_1 + ta_3$$

Now we substitute the coordinates of the line (x,y,z) in the plane:

$$A(x_1+ta_1)+B(y_1+ta_2)+C(z_1+ta_3)+D=0$$

$$(Aa_1+Ba_2+Ca_3)t+Ax_1+By_1+Cz_1+D=0$$

Now let $$a=Aa_1+Ba_2+Ca_3$$ and

$$b=Ax_1+By_1+Cz_1+D$$.

we got $$at+b=0$$

If a≠0, t=-b/a

So the point where the line intersects the plane is:

$$M(x_1 - \frac{b}{a}a_1 , y_1 - \frac{b}{a}a_2 , z_1 - \frac{b}{a}a_3)$$

Regards.

 

[tiny-tim]

Welcome to PF!

Homework Statement

Find the point as which the line intersects the given plane.

Homework Equations

Line: x = y - 1 = 2z
Plane: 4x - y + 3z = 8

The Attempt at a Solution

I understand how to use the cross product, dot product, and find intercepts. This problem is in section 13.5 #45 of the James Stewart Calculus book. I understand the idea but need some help in solving the problem. Thanks!

Hi jdj333! Welcome to PF!

You don't need cross and dot products for this!

Hint: finding an intersection is just a simultaneous equations problem …

just substitute the line equation into the plane equation (to make it all y, say), and solve.