1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
grigory [225]
3 years ago
7

Find two vectors in R2 with Euclidian Norm 1 whoseEuclidian inner product with (3,1) is zero.

Mathematics
1 answer:
alina1380 [7]3 years ago
5 0

Answer:

v_1=(\frac{1}{10},-\frac{3}{10})

v_2=(-\frac{1}{10},\frac{3}{10})

Step-by-step explanation:

First we define two generic vectors in our \mathbb{R}^2 space:

  1. v_1 = (x_1,y_1)
  2. v_2 = (x_2,y_2)

By definition we know that Euclidean norm on an 2-dimensional Euclidean space \mathbb{R}^2 is:

\left \| v \right \|= \sqrt{x^2+y^2}

Also we know that the inner product in \mathbb{R}^2 space is defined as:

v_1 \bullet v_2 = (x_1,y_1) \bullet(x_2,y_2)= x_1x_2+y_1y_2

So as first condition we have that both two vectors have Euclidian Norm 1, that is:

\left \| v_1 \right \|= \sqrt{x^2+y^2}=1

and

\left \| v_2 \right \|= \sqrt{x^2+y^2}=1

As second condition we have that:

v_1 \bullet (3,1) = (x_1,y_1) \bullet(3,1)= 3x_1+y_1=0

v_2 \bullet (3,1) = (x_2,y_2) \bullet(3,1)= 3x_2+y_2=0

Which is the same:

y_1=-3x_1\\y_2=-3x_2

Replacing the second condition on the first condition we have:

\sqrt{x_1^2+y_1^2}=1 \\\left | x_1^2+y_1^2 \right |=1 \\\left | x_1^2+(-3x_1)^2 \right |=1 \\\left | x_1^2+9x_1^2 \right |=1 \\\left | 10x_1^2 \right |=1 \\x_1^2= \frac{1}{10}

Since x_1^2= \frac{1}{10} we have two posible solutions, x_1=\frac{1}{10} or x_1=-\frac{1}{10}. If we choose x_1=\frac{1}{10}, we can choose next the other solution for x_2.

Remembering,

y_1=-3x_1\\y_2=-3x_2

The two vectors we are looking for are:

v_1=(\frac{1}{10},-\frac{3}{10})\\v_2=(-\frac{1}{10},\frac{3}{10})

You might be interested in
A bookshelf is 42 inches long. How many books of length 1\frac{1}{2}1 2 1 ​ inches will fit on the bookshelf? Explain your reaso
Blababa [14]

Answer: See explanation

Step-by-step explanation:

Your question isn't well written but I saw an identical question:

A bookshelf is 42 inches long. Each book is 1 1/4 inches wide. How many books will fit on the shelf?

For us to solve this, we would divide the length of the bookshelf by the width of each book. This would be:

= 42 ÷ 1 1/4

= 42 ÷ 5/4

= 42 × 4/5

= 33 3/5

Therefore, the number of books that will fit on the shelf is 33.

7 0
2 years ago
Can someone help me with this question?
vova2212 [387]
3 feet because you multiply 0.2 and 15
7 0
3 years ago
Read 2 more answers
What is the completely factored form of 12xy-9x-8y+6
boyakko [2]

Answer: (4y - 3)(3x - 2)

Step-by-step explanation: Factor the polynomial.

Hope this helps! :) ~Zane

5 0
3 years ago
Read 2 more answers
An inequality is shown below.
Stells [14]
B.
You are looking for something that does not let 25x go over 80.
8 0
3 years ago
The angle measure in degrees that corresponds to 7/10ths of a circle is what?
Serggg [28]
7/10 of the degrees of a circle is 252 . I dont know the second part of the question sorry.
8 0
3 years ago
Other questions:
  • mr. williams physical education class lasts 7/8 hour. how many minutes are spent warming up and cooling down if they spent 3/10?
    12·1 answer
  • James has a piece of constrution paper with area of 65 1/4 inches . It is 9 2/3 inches long. What is the width of the piece of c
    13·1 answer
  • Use the substitution method to solve the following system of equations: 3x + 5y = 3 x + 2y = 0 (1 point) (1, 1) (6, −3) (6, 1) (
    12·1 answer
  • **Marking Brainliest** Which of the following are point-slope equations of the line going through (-4, 6) and (1, 2)? Check all
    9·2 answers
  • Find the surface area of the following triangular prisms
    13·1 answer
  • Mr. Mutler posts his students' artwork on a bulletin board.
    9·1 answer
  • Miriam is going to the county fair. Each ride ticket costs $2 and admission costs $7. Miriam has $25 to spend at the fair.
    9·1 answer
  • Draw and label a coordinate plane, including the four quadrants and the positive or negative values of x and y in each quadrant.
    11·2 answers
  • Simplify the expression 111,000*0.072 using scientific notation and express your answer in scientific notation.
    12·1 answer
  • What is the right answer <br> giving out brainliest
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!