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4 years ago

5

Note: You may use a calculator to solve the following problems.

1 answer:

Verizon [17]4 years ago

4 0

Use a calculator it will help.

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kenny6666 [7]

Answer:

The value of the constant $k$ so that $\vec u_{3}$ is a linear combination of $\vec u_{1}$ and $\vec u_{2}$ is $\frac{7}{10}$ .

Step-by-step explanation:

Let be $\vec u_{1} = [2,3,1]$ , $\vec u_{2} = [4,1,0]$ and $\vec u_{3} = [1, 2,k]$ , $\vec u_{3}$ is a linear combination of $\vec u_{1}$ and $\vec u_{3}$ if and only if:

$\alpha_{1} \cdot \vec u_{1} + \alpha_{2} \cdot \vec u_{2} +\alpha_{3}\cdot \vec u_{3} = \vec O$ (Eq. 1)

Where:

$\alpha_{1}$ , $\alpha_{2}$ , $\alpha_{3}$ - Scalar coefficients of linear combination, dimensionless.

By dividing each term by $\alpha_{3}$ :

$\lambda_{1}\cdot \vec u_{1} + \lambda_{2}\cdot \vec u_{3} = -\vec u_{3}$

$\vec u_{3}=-\lambda_{1}\cdot \vec u_{1}-\lambda_{2}\cdot \vec u_{2}$ (Eq. 2)

$\vec O$ - Zero vector, dimensionless.

And all vectors are linearly independent, meaning that at least one coefficient must be different from zero. Now we expand (Eq. 2) by direct substitution and simplify the resulting expression:

$[1,2,k] = -\lambda_{1}\cdot [2,3,1]-\lambda_{2}\cdot [4,1,0]$

$[1,2,k] = [-2\cdot\lambda_{1},-3\cdot \lambda_{1},-\lambda_{1}]+[-4\cdot \lambda_{2},-\lambda_{2},0]$

$[0,0,0] = [-2\cdot \lambda_{1},-3\cdot \lambda_{1},-\lambda_{1}]+[-4\cdot \lambda_{2},-\lambda_{2},0]+[-1,-2,-k]$

$[-2\cdot \lambda_{1}-4\cdot \lambda_{2}-1,-3\cdot \lambda_{1}-\lambda_{2}-2,-\lambda_{1}-k] =[0,0,0]$

The following system of linear equations is obtained:

$-2\cdot \lambda_{1}-4\cdot \lambda_{2}= 1$ (Eq. 3)

$-3\cdot \lambda_{1}-\lambda_{2}= 2$ (Eq. 4)

$-\lambda_{1}-k = 0$ (Eq. 5)

The solution of this system is:

$\lambda_{1} = -\frac{7}{10}$ , $\lambda_{2} = \frac{1}{10}$ , $k = \frac{7}{10}$

The value of the constant $k$ so that $\vec u_{3}$ is a linear combination of $\vec u_{1}$ and $\vec u_{2}$ is $\frac{7}{10}$ .

4 0

4 years ago

Write the expression 12^-2 in simplest form.

olga2289 [7]

Answer:

Step-by-step explanation:

1/12^2 = 1/144

8 0

2 years ago

If y varies directly with x, find the constant variation with x= 4 and y= -26

kondor19780726 [428]

The constant variation is k=-13/2

6 0

4 years ago

Read 2 more answers

I beg someone to answer ive posted this 3 times and im still not sure how to do it.....

Anni [7]

Answer:

D) 3 X 10⁷

Step-by-step explanation:

<u>Estimated days:</u>

Jan 2000 to June 2001 = 1.5 years = 365*1.5 days ≈ 548 days

1 day = 24 hours = 24*60 minutes = 24*60*60 seconds =

86400 seconds

548 days = 548 * 86400 seconds =

47347200 ≈ 4.7*10⁷

Best estimate is D) 3 X 10⁷

8 0

3 years ago

katovenus [111]

Answer:

(-4,2)

Step-by-step explanation:

Kindly check the attatched image to see the system graphed

The two equations intersect at (-4,2) so the solution is (-4,2)

7 0

2 years ago