consider the plane in $\mathbb{r}^3$ that passes through the point $p(-2, 3, 0)$ and has normal vector $\mathbf{n} = (2, 3, 1)$.

find the scalar equation of this plane.

find the general form of the equation of this plane.

Question

consider the plane in $\mathbb{r}^3$ that passes through the point $p(-2, 3, 0)$ and has normal vector $\mathbf{n} = (2, 3, 1)$.

find the scalar equation of this plane.

find the general form of the equation of this plane.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Equation of a Plane", "Normal Vector", "Scalar Equation" ], "current_concepts": [ "Equation of a Plane", "Normal Vector", "Scalar Equation", "General Form Equation" ] } ## Set up the scalar equation of the plane $$ \begin{aligned} &P(x_0, y_0, z_0) = (-2, 3, 0) \ &\mathbf{n} = (a, b, c) = (2, 3, 1) \ &a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \ &2(x - (-2)) + 3(y - 3) + 1(z - 0) = 0 \ &2(x + 2) + 3(y - 3) + z = 0 \end{aligned} $$ ## Expand to find the general form of the equation $$ \begin{aligned} &2x + 4 + 3y - 9 + z = 0 \ &2x + 3y + z - 5 = 0 \end{aligned} $$ ## Question 1 $2(x + 2) + 3(y - 3) + z = 0$ ## Question 2 $2x + 3y + z - 5 = 0$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Equation of a Plane" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the scalar equation of the plane

Expand to find the general form of the equation

Answer:

Question 1

Question 2