Chapter 11: Q. 2 (page 879)

Equation of a plane: Find the equation of the plane determined by the vectors (1, −3, 5)and (2, 4, −1) and containing the point(0, 3, −2).

Short Answer

Expert verified

The equation of the plane is $- 17 x + 11 y + 10 z - 13 = 0 .$

Step by step solution

Step 1. Given Information.

The given vectors of the plane are (1, −3, 5)and (2, 4, −1)and the given containing point is(0, 3, −2).

Step 2. Finding the equation of a plane.

To find the equation of a plane determined by the vectors and containing the point, we will use the formula: $(\vec{r} - \overset{⇀}{a}) \cdot (\vec{α} \times \vec{β}) = 0 .$

Let's find the cross product,

role="math" localid="1649641070012" $(\vec{α} \times \vec{β}) = |\begin{matrix} i & j & k \\ 1 & - 3 & 5 \\ 2 & 4 & - 1 \end{matrix}| = i ((- 3) (- 1) - (4) (5)) - j ((1) (- 1) - (2) (5)) + k ((1) (4) - (2) (- 3)) = - 17 i + 11 j + 10 k$

Now, let's do the difference,

role="math" localid="1649641452589" $(\vec{r} - \vec{a}) = (x i + y j + z k) - (0 i + 3 j - 2 k) = i (x - 0) + j (y - 3) + k (z - (- 2)) = x i + (y - 3) j + (z + 2) k$

Step 3. Calculate.

Now, let's do the dot product,

$[x i + (y - 3) j + (z + 2) k] \cdot (- 17 i + 11 j + 10 k) = 0 (x) (- 17) + (y - 3) (11) + (z + 2) (10) = 0 - 17 x + 11 y - 33 + 10 z + 20 = 0 - 17 x + 11 y + 10 z - 13 = 0$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Equation of a plane: Find the equation of the plane determined by the vectors (1, −3, 5)and (2, 4, −1) and containing the point(0, 3, −2).

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the equation of a plane.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.