If the equation of a line is

𝑥
=
[
1

2

3
]
+
𝑡
[
1

0

−
1
]

Find the value of t that results in an intersection with the plane

6x+5y+4z=1

Enter your answer to 1 decimal place if it is not a whole number.

Question

If the equation of a line is

𝑥
=
[
1

2

3
]
+
𝑡
[
1

0

−
1
]

Find the value of t that results in an intersection with the plane

6x+5y+4z=1

Enter your answer to 1 decimal place if it is not a whole number.

BlackTom AI · Accepted Answer

Parameterize the line: x = 1 + t, y = 2, z = 3 − t.

1. Relevant knowledge points and concepts
- A line in parametric form gives coordinates (x,y,z) as functions of parameter t.
- A plane given by ax + by + cz = d is satisfied by any point (x,y,z) on that plane.
- Intersection of a line and a plane is found by substituting the parametric coordinates into the plane equation and solving for t.
- If the direction vector of the line is orthogonal to the plane normal, the line is parallel to the plane; otherwise there is at most one intersection point.

2. Complete reasoning process step by step
Substitute x, y, z from the line into the plane equation 6x + 5y + 4z = 1 and solve for t.

3. Calculations with all intermediate steps
Start with substitution:
6(1 + t) + 5(2) + 4(3 − t) = 1

Expand each term:
6 + 6t + 10 + 12 − 4t = 1

Combine like terms (constants and t-terms):
(6 + 10 + 12) + (6t − 4t) = 1
28 + 2t = 1

Isolate t:
2t = 1 − 28
2t = −27
t = −27 / 2
t = −13.5

4. Why the correct answer is correct
The algebraic substitution reduces the geometric intersection condition to a single linear equation in t. Solving that equation yields t = −13.5, the unique parameter value for which the point on the line satisfies the plane equation. Also, the direction vector v = (1,0,−1) has dot product with the plane normal n = (6,5,4) equal to 6 − 4 = 2 ≠ 0, so the line is not parallel and there is exactly one intersection.

5. Why other potential answers are incorrect
No other t can satisfy the linear equation 28 + 2t = 1, so any different numeric value would fail the plane equation. There is a unique solution because the line is not parallel to the plane.

6. Conclusion
The value of t that gives the intersection is t = −13.5, which matches the provided answer.

If the equation of a line is 𝑥 = [ 1 2 3 ] + 𝑡 [ 1 0 − 1 ] Find the value of t that results in an intersection with the plane 6x+5y+4z=1 Enter your answer to 1 decimal place if it is not a whole number.简答题

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