5x-5y+4z=15 3x+5y-6z=7 -x+5y-5z=20

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

I need an answer ASAP with an explanation please!

Butoxors [25]

Answer:

y intercept (0;4)

Step-by-step explanation:

let x = 0 because the graph will intersect the y-axis at the value of 0 for the x-axis

3 0

3 years ago

The mean salary of people living in a certain city is $37500 with a standard deviation of $1561. a sample of 33 people is select

Murrr4er [49]

The probability is 0.9671.

The z-score for this is given by

z = (x - μ)/(σ/√n)
z = (38000-37500)/(1561/(√33)) = 1.84.

Using a z-table (http://www.z-table.com) we see that the area under the curve to the left of, less than, this is 0.9671.

8 0

3 years ago

Write the coordinate rule to describe The translasion

sergejj [24]

Answer:

The coordinates are:

A ( 4,7 )

B ( 8,10 )

c ( 7,7 )

Step-by-step explanation:

You could take the original coordinates and add the x value with 3 and y value with 6. Like for example...

The original coordinates of C is ( 4,1 ). Hence you could add the x value with 3 and y value with 6. 4 + 3 and 1 + 6 giving you the coordinates after translation = ( 7,7 )

6 0

2 years ago

PLZ HELP !!!!!! IT’S DUE TODAY

Kay [80]

Answer:

jvhhhhchgcgvvv

Step-by-step explanation:

4 0

3 years ago

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