Im not sure what to do here or how to start for AP calculus

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

Samantha received an allowance of $3 every week. By babysitting, she earned $30 every week. How much did Samantha have in four w

kvasek [131]

Okay, so there are 4 weeks in a month. 30x4 is 120,and 3x4 is 12. Add the 2 together and you have 132$

6 0

3 years ago

you and a friend go out to dinner. The cost of the meal is $45.22. The sales tax is 5% and deliver 16% tip. what is the total co

kumpel [21]

Answer: $57.72

Step-by-step explanation:

4 0

3 years ago

Can someone clearly explain and workout Q19? Much appreciated

Anna007 [38]

The x coefficient is 2
Take half of 2 to get 2/2 = 1
Square 1 to get 1^2 = 1

Add and subtract 1 to the expression. This is the same as adding 0.

x^2 + 2x + 2
x^2 + 2x + 2 + 0
x^2 + 2x + 2 + (1-1)
(x^2 + 2x + 1) + 2 - 1
(x+1)^2 + 1

Answer: (x+1)^2 + 1

5 0

3 years ago

There are between 50 & 60 eggs in a basket. When roza counts by 3's there are 2 leftover and when she counts by 5's there ar

avanturin [10]

There are 59 eggs in the basket

Step-by-step explanation:

There are between 50 & 60 eggs in a basket

When Roza counts by 3's there are 2 leftover
When she counts by 5's there are 4 leftover

We need to find how many eggs in the basket

∵ When Roza counts by 3's there are 2 leftover

∵ When she counts by 5's there are 4 leftover

- That means the number of eggs when divided by 3 give

remainder 2 and when divided by 5 give remainder 4

- To find the number look for the numbers that divisible by 5 and

add 4 to them and then check the remainder when they divided

by 3

∵ Between 50 and 60 there is only one number divisible by 5

is 55 (not including 50 or 60)

∵ 55 + 4 = 59

∴ 59 gives remainder 4 when divided by 5

Let us check them with 3

∵ 59 - 2 = 57

∵ 57 is divisible by 3 ⇒ (57 ÷ 3 = 19)

∴ 59 when divide by 3 gives remainder 2

∴ 59 gives remainder 4 when divided by 5 and remainder 2 when

divided by 3

∴ The number of eggs in the basket is 59

There are 59 eggs in the basket

Learn more:

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6 0

3 years ago