When writing mathematics, make sure that you use the correct notations for superscripts, subscripts and number systems. We use blackboard letters for number systems. In , the command \mathbb will create a blackboard letter, e.g., \mathbb{R} for the set of all real numbers .
Choose the options which you think are the best from the list below. Note that each sentence is trying to express the same information.

Question

BlackTom AI · Accepted Answer

A general plane in ℝ^3 is represented by ax_1 + bx_2 + cx_3 = d, with a^2 + b^2 + c^2 ≠ 0.

1. Relevant knowledge points and concepts
- A plane in three-dimensional Euclidean space ℝ^3 can be written in linear form as a x_1 + b x_2 + c x_3 = d.
- The vector n = (a, b, c) is a normal vector to the plane; its components are the coefficients of x_1, x_2, x_3.
- The condition a^2 + b^2 + c^2 ≠ 0 guarantees that the normal vector is not the zero vector (i.e., at least one coefficient is nonzero), so the equation indeed defines a plane rather than a degenerate statement.

2. Step-by-step reasoning
- Start with the dot product representation: n · x = d, where n = (a,b,c) and x = (x_1,x_2,x_3).
- Expanding the dot product gives a x_1 + b x_2 + c x_3 = d.
- If n = (0,0,0) then a = b = c = 0 and the left-hand side is 0 for all x, so the equation becomes 0 = d. If d ≠ 0 this is impossible (no solution); if d = 0 it is tautologically true (all of ℝ^3), so neither case describes a single plane. Hence require a^2 + b^2 + c^2 ≠ 0.

3. Intermediate/calculation steps
- n · x = a x_1 + b x_2 + c x_3.
- Norm squared: ||n||^2 = a^2 + b^2 + c^2; require ||n||^2 ≠ 0.

4. Why the correct answer is correct
- Option d uses correct standard notation: ℝ^3 (blackboard bold for the number system), proper subscripts x_1, x_2, x_3, superscripts for squares a^2, etc., and the precise inequality a^2 + b^2 + c^2 ≠ 0 which enforces a nonzero normal vector.

5. Why the other options are incorrect
- Other choices either fail to use standard mathematical notation (e.g., writing x1 instead of x_1 or using plain text caret notation inconsistently), or display the inequality or number-system notation less precisely. Such formatting mistakes make them less correct or less standard than option d.

6. Conclusion
- Therefore the correct, properly notated statement is option d: "A general equation of a plane in ℝ^3 is given by ax_1 + bx_2 + cx_3 = d, where a^2 + b^2 + c^2 ≠ 0." This matches the provided answer.

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