What are their perimeters?

Find an example for each of vectors x, y ∈ V in R.

rjkz [21]

(a) Both conditions are satisfied with x = (1, 0) for $\mathbb R^2$ and x = (1, 0, 0) for $\mathbb R^3$ :

||(1, 0)|| = √(1² + 0²) = 1

max{1, 0} = 1

||(1, 0, 0)|| = √(1² + 0² + 0²) = 1

max{1, 0, 0} = 1

(b) This is the well-known triangle inequality. Equality holds if one of x or y is the zero vector, or if x = y. For example, in $\mathbb R^2$ , take x = (0, 0) and y = (1, 1). Then

||x + y|| = ||(0, 0) + (1, 1)|| = ||(1, 1)|| = √(1² + 1²) = √2

||x|| + ||y|| = ||(0, 0)|| + ||(1, 1)|| = √(0² + 0²) + √(1² + 1²) = √2

The left side is strictly smaller if both vectors are non-zero and not equal. For example, if x = (1, 0) and y = (0, 1), then

||x + y|| = ||(1, 0) + (0, 1)|| = ||(1, 1)|| = √(1² + 1²) = √2

||x|| + ||y|| = ||(1, 0)|| + ||(0, 1)|| = √(1² + 0²) + √(0² + 1²) = 2

and of course √2 < 2.

Similarly, in $\mathbb R^3$ you can use x = (0, 0, 0) and y = (1, 1, 1) for the equality, and x = (1, 0, 0) and y = (0, 1, 0) for the inequality.

(c) Recall the dot product identity,

x • y = ||x|| ||y|| cos(θ),

where θ is the angle between the vectors x and y. Both sides are scalar, so taking the norm gives

||x • y|| = ||(||x|| ||y|| cos(θ)|| = ||x|| ||y|| |cos(θ)|

Suppose x = (0, 0) and y = (1, 1). Then

||x • y|| = |(0, 0) • (1, 1)| = 0

||x|| • ||y|| = ||(0, 0)|| • ||(1, 1)|| = 0 • √2 = 0

For the inequality, recall that cos(θ) is bounded between -1 and 1, so 0 ≤ |cos(θ)| ≤ 1, with |cos(θ)| = 0 if x and y are perpendicular to one another, and |cos(θ)| = 1 if x and y are (anti-)parallel. You get everything in between for any acute angle θ. So take x = (1, 0) and y = (1, 1). Then

||x • y|| = |(1, 0) • (1, 1)| = |1| = 1

||x|| • ||y|| = ||(1, 0)|| • ||(1, 1)|| = 1 • √2 = √2

In $\mathbb R^3$ , you can use the vectors x = (1, 0, 0) and y = (1, 1, 1).

8 0

3 years ago

How find a partial quotients to divide 6 and 258

34kurt

Well look u can trust me at this im really good at it the answer is 43 trust me

6 0

3 years ago

Q # 15 I need help please to resolve

kow [346]

The sequence is

740, 707.60,675.20,642.80,...

Here first term is 740, and common difference, d is

$707.60-740 = -32.40$

Now we use the formula of nth term, which is

$a_{n} = a+(n-1)d$

Substituting the values of a,n and d, we will get

$=740+(n-1)(-32.40), 23 months$

Correct option is the second option .

7 0

4 years ago

What is 4(x−2)=3(2y−1) in standard form?

exis [7]

Answer: 4x-6y=5

Step-by-step explanation: Using the formula, Ax+By=C, write the equation in standard form.

Hope this helps you out! ☺

8 0

3 years ago

6x+12y+5+2y+8 what’s the coefficient of y and what’s the constant

Rudik [331]

The coefficient of y is 14 and the constant is 13.

You can find this by combining like terms first.

6x + 12y + 5 + 2y + 8

6x + 14y + 5 + 8

6x + 14y + 13

Now the coefficient is the number next to y. The constant is the number on its own.

4 0

3 years ago