Chapter 3: Q17P (page 105)

Find three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

Short Answer

Expert verified

The three vectors are $2 \hat{i} + \hat{j} - \hat{k}$ , $\hat{i} + 3 \hat{j} + 5 \hat{k}$ and $3 \hat{i} + 4 \hat{j} + 4 \hat{k}$ .

Step by step solution

Required

Three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

Results used

Vector $\vec{A} \times \vec{B}$ is perpendicular to vector $\vec{A}$ as well as $\vec{B}$ .

Consider two vectors and find a perpendicular vector

Consider two vectors,

$\vec{A} = 2 \hat{i} + \hat{j} - \hat{k}$

$\vec{B} = \hat{i} + 3 \hat{j} - 2 \hat{k}$

$\vec{A} \times \vec{B} = |\begin{matrix} i & j & k \\ 2 & 1 & - 1 \\ 1 & 3 & - 2 \end{matrix}| = (- 2 + 3) \hat{i} - (- 4 + 1) \hat{j} + (6 - 1) \hat{k} = \hat{i} + 3 \hat{j} + 5 \hat{k}$

So, $\vec{A}$ and $\vec{A} \times \vec{B}$ are perpendicular vectors.

Hence, these are 2 perpendicular sides of required triangle.

Find third side of triangle

Two sides of triangle are already known.

Third side can be found using vector law of addition.

Third vector is

$\vec{A} + (\vec{A} \times \vec{B}) = (2 \hat{i} + \hat{j} - \hat{k}) + (\hat{i} + 3 \hat{j} + 5 \hat{k}) = (2 + 1) \hat{i} + (1 + 3) \hat{j} + (- 1 + 5) \hat{k} = 3 \hat{i} + 4 \hat{j} + 4 \hat{k}$

Conclusion

First side is $2 \hat{i} + \hat{j} - \hat{k}$ .

Second side perpendicular to first one is $\hat{i} + 3 \hat{j} + 5 \hat{k}$

Hypotenuse is $3 \hat{i} + 4 \hat{j} + 4 \hat{k}$ .

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 Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

Short Answer

Step by step solution

Required

Results used

Consider two vectors and find a perpendicular vector

Find third side of triangle

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Waves Physics

Space Physics

Physics of Motion

Turning Points in Physics

Mechanics and Materials

Study anywhere. Anytime. Across all devices.