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]
2 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]2 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
Find the value of x
geniusboy [140]

-1, maybe or maybe not this is my first time doing this

7 0
3 years ago
PLEASE HELP!!!!!!!!!
Naddik [55]
The answer is A. The player runs 7 meters forward and then runs 7 meters in the opposite direction. Hope it helped!

- Shadow.
4 0
3 years ago
Zachary bought 8 hot dogs at the fair. He paid a total cost of $12.00 for the food. Find the unit cost of one hot dog.
masya89 [10]

Answer:.66

Step-by-step explanation:

You divide 8 and 12

3 0
3 years ago
Read 2 more answers
At 2 hours before sunset, the temperature is 0°F. At 2 hours after sunset, the temperature is −4°F. Identify the slope, y-interc
Liula [17]

Answer

slope: 1

y-intercept: 0,–2

x-intercept: -2,0

Step-by-step explanation

I just did this question

7 0
3 years ago
Simplify Exponents (X^3)^5
lbvjy [14]

Answer:

x ^15

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
Other questions:
  • A bag contains 5 purple, 24 green, 12 yellow, and 18 gray marbles. A marble is drawn at random. What is the theoretical probabil
    8·2 answers
  • What would the answer to -|-24| be?
    11·1 answer
  • Suppose you are going to graph the data in the table below. What data should be represented on each axis, and what would be the
    9·2 answers
  • given a and b are numbers,and a+b =180,which statement also must be true ? a=180-b , a-180=b , 360=2a+2b, a=90 and B=90
    5·1 answer
  • What is the prescribed dosage of a certain medicine for a 6-year-old child if the adult dosage of
    13·1 answer
  • LaPeach Elementary School uses 4000 pencils each year. The school has 500 students. What is the unit rate per student?
    11·2 answers
  • PLS HELP FIND THE AREA OF THE SHAPE BELOW
    10·2 answers
  • What’s the answer please help
    6·1 answer
  • Find an ordered pair (x, y) that is a solution to the equation.<br> 3x-y=5
    6·1 answer
  • Given​ f(x) = x2−2x+2​, find the​ value(s) for x such that ​f(x)=37.
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!