Chapter 9: Q. 54 (page 603)

Find a unit vector normal to the plane containing
$\vec{v} = 2 \hat{i} + 3 \hat{j} - \hat{k} \vec{w} = - 2 \hat{i} - 4 \hat{j} + 3 \hat{k}$

Short Answer

Expert verified

$\hat{a} = \frac{- 13}{\sqrt{237}} \hat{i} + \frac{8}{\sqrt{237}} \hat{j} - \frac{2}{\sqrt{237}} \hat{k}$

Step by step solution

Step 1. Given vectors in the plane

$\vec{v} = 2 \hat{i} + 3 \hat{j} - \hat{k} \vec{w} = - 2 \hat{i} - 4 \hat{j} + 3 \hat{k}$

Step 2. The cross product

We know that the unit vector perpendicular to

is:

$\hat{a} = \pm \frac{\vec{v} \times \vec{w}}{|\vec{v} \times \vec{w}|}$

So;

$\vec{v} = 2 \hat{i} + 3 \hat{j} - \hat{k} \vec{w} = - 2 \hat{i} - 4 \hat{j} - 3 \hat{k} \vec{v} \times \vec{w} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & - 1 \\ - 2 & - 4 & - 3 \end{matrix}| \vec{v} \times \vec{w} = \hat{i} (- 9 - 4) - \hat{j} (- 6 - 2) + \hat{k} (- 8 + 6) \vec{v} \times \vec{w} = - 13 \hat{i} + 8 \hat{j} - 2 \hat{k} | \vec{v} \times \vec{w} | = \sqrt{169 + 4 + 64} | \vec{v} \times \vec{w} | = \sqrt{237}$

Step 3. The unit vector is :

$\hat{a} = \frac{- 13}{\sqrt{237}} \hat{i} + \frac{8}{\sqrt{237}} \hat{j} - \frac{2}{\sqrt{237}} \hat{k}$

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.

Find a unit vector normal to the plane containing
$\vec{v} = 2 \hat{i} + 3 \hat{j} - \hat{k} \vec{w} = - 2 \hat{i} - 4 \hat{j} + 3 \hat{k}$

Short Answer

Step by step solution

Step 1. Given vectors in the plane

Step 2. The cross product

Step 3. The unit vector is :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Logic and Functions

Probability and Statistics

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find a unit vector normal to the plane containingv→=2i^+3j^-k^w→=-2i^-4j^+3k^

Short Answer

Step by step solution

Step 1. Given vectors in the plane

Step 2. The cross product

Step 3. The unit vector is :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Logic and Functions

Probability and Statistics

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find a unit vector normal to the plane containing
$\vec{v} = 2 \hat{i} + 3 \hat{j} - \hat{k} \vec{w} = - 2 \hat{i} - 4 \hat{j} + 3 \hat{k}$