Point on the Line?

Question

a, is the point (11,14,17,18) on the line (1,2,3,4)+t(5,6,7,8)?

b, does the point (4,3,2) lie in the plane (1,2,3)+t(1,1,0)+s(0,1,1)?

Answer 1

(11,14,17,18) on the line (1,2,3,4)+t(5,6,7,8)
11 = 1 + 5t
t= 2

(1,2,3,4)+2(5,6,7,8) = (1,2,3,4)+(10,12,14,16) = (11,14,17,20)

No because in the fourth number inside the parenthesis is 18, but it's not equal to 20.

(4,3,2) lie in the plane (1,2,3)+t(1,1,0)+s(0,1,1)

4 = 1 + t
t= 3

2 = 3 + s
s= -1

(1,2,3)+3(1,1,0)+(-1)(0,1,1) = (1,2,3)+(3,3,0)+(0,-1,-1) = (4,4,2)

no since in the second number, it is 4 but it is 3.

Answer 2

a) Does the point P(11,14,17,18) on the line
L(t) = <1,2,3,4> + t<5,6,7,8>?

Solve for t.

x = 1 + 5t = 11
5t = 10
t = 2

<1,2,3,4> + 2<5,6,7,8> = <11, 14, 17, 20>

No, the point (11,14,17,18) does NOT lie on the line. The solution t = 2 is inconsistent for the last variable.
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b) Does the point (4,3,2) lie in the plane
(1,2,3) + t(1,1,0) + s(0,1,1)?

The normal vector n, to the plane will be orthogonal to both directional vectors of the plane. Take the cross product.

n = <1, 1, 0> X <0, 1, 1> = <1, -1, 1>

With the normal vector n and a point on the plane (1, 2, 3), we can write the equation of the plane.

1(x - 1) - 1(y - 2) + 1(z - 3) = 0
x - 1 - y + 2 + z - 3 = 0
x - y + z - 2 = 0

Now plug in the point (4, 3, 2) to see if it is in the plane.

4 - 3 + 2 - 2 = 1 ≠ 0

So the point (4, 3, 2) is NOT in the plane.