Question
Prove that the line {p:p = (1, 3, -1) + t(0, 3, 5)} lies entirely in the plane
{(x, y, z): 2x – 5y + 3z = -16}
{(x, y, z): 2x – 5y + 3z = -16}
Answers
GPT 3.5
To prove that the line lies entirely in the plane, we need to show that every point on the line satisfies the equation of the plane.
Given the line {p: p = (1, 3, -1) + t(0, 3, 5)} and the plane {(x, y, z): 2x - 5y + 3z = -16}, we want to show that for any value of t, the point p = (1, 3, -1) + t(0, 3, 5) satisfies the equation 2x - 5y + 3z = -16.
Let's substitute the values of x, y, and z using the equation for the line:
x = 1 + 0t = 1
y = 3 + 3t
z = -1 + 5t
Now, let's substitute these values into the equation of the plane:
2(1) - 5(3 + 3t) + 3(-1 + 5t) = -16
2 - 15 - 15t - 3 + 15t = -16
-16 = -16
As we can see, the equation holds true for any value of t. Therefore, every point on the line lies in the plane. Hence, the line lies entirely in the plane {(x, y, z): 2x - 5y + 3z = -16}.
Given the line {p: p = (1, 3, -1) + t(0, 3, 5)} and the plane {(x, y, z): 2x - 5y + 3z = -16}, we want to show that for any value of t, the point p = (1, 3, -1) + t(0, 3, 5) satisfies the equation 2x - 5y + 3z = -16.
Let's substitute the values of x, y, and z using the equation for the line:
x = 1 + 0t = 1
y = 3 + 3t
z = -1 + 5t
Now, let's substitute these values into the equation of the plane:
2(1) - 5(3 + 3t) + 3(-1 + 5t) = -16
2 - 15 - 15t - 3 + 15t = -16
-16 = -16
As we can see, the equation holds true for any value of t. Therefore, every point on the line lies in the plane. Hence, the line lies entirely in the plane {(x, y, z): 2x - 5y + 3z = -16}.