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
vodka [1.7K]
3 years ago
10

Match each vector operation with its resultant vector expressed as a linear combination of the unit vectors i and j.

Mathematics
1 answer:
Cloud [144]3 years ago
7 0

Answer:

3u - 2v + w = 69i + 19j.

8u - 6v = 184i + 60j.

7v - 4w = -128i + 62j.

u - 5w = -9i + 37j.

Step-by-step explanation:

Note that there are multiple ways to denote a vector. For example, vector u can be written either in bold typeface "u" or with an arrow above it \vec{u}. This explanation uses both representations.

\displaystyle \vec{u} = \langle 11, 12\rangle =\left(\begin{array}{c}11 \\12\end{array}\right).

\displaystyle \vec{v} = \langle -16, 6\rangle= \left(\begin{array}{c}-16 \\6\end{array}\right).

\displaystyle \vec{w} = \langle 4, -5\rangle=\left(\begin{array}{c}4 \\-5\end{array}\right).

There are two components in each of the three vectors. For example, in vector u, the first component is 11 and the second is 12. When multiplying a vector with a constant, multiply each component by the constant. For example,

3\;\vec{v} = 3\;\left(\begin{array}{c}11 \\12\end{array}\right) = \left(\begin{array}{c}3\times 11 \\3 \times 12\end{array}\right) = \left(\begin{array}{c}33 \\36\end{array}\right).

So is the case when the constant is negative:

-2\;\vec{v} = (-2)\; \left(\begin{array}{c}-16 \\6\end{array}\right) =\left(\begin{array}{c}(-2) \times (-16) \\(-2)\times(-6)\end{array}\right) = \left(\begin{array}{c}32 \\12\end{array}\right).

When adding two vectors, add the corresponding components (this phrase comes from Wolfram Mathworld) of each vector. In other words, add the number on the same row to each other. For example, when adding 3u to (-2)v,

3\;\vec{u} + (-2)\;\vec{v} = \left(\begin{array}{c}33 \\36\end{array}\right) + \left(\begin{array}{c}32 \\12\end{array}\right) = \left(\begin{array}{c}33 + 32 \\36+12\end{array}\right) = \left(\begin{array}{c}65\\48\end{array}\right).

Apply the two rules for the four vector operations.

<h3>1.</h3>

\displaystyle \begin{aligned}3\;\vec{u} - 2\;\vec{v} + \vec{w} &= 3\;\left(\begin{array}{c}11 \\12\end{array}\right) + (-2)\;\left(\begin{array}{c}-16 \\6\end{array}\right) + \left(\begin{array}{c}4 \\-5\end{array}\right)\\&= \left(\begin{array}{c}3\times 11 + (-2)\times (-16) + 4\\ 3\times 12 + (-2)\times 6 + (-5) \end{array}\right)\\&=\left(\begin{array}{c}69\\19\end{array}\right) = \langle 69, 19\rangle\end{aligned}

Rewrite this vector as a linear combination of two unit vectors. The first component 69 will be the coefficient in front of the first unit vector, i. The second component 19 will be the coefficient in front of the second unit vector, j.

\displaystyle \left(\begin{array}{c}69\\19\end{array}\right) = \langle 69, 19\rangle = 69\;\vec{i} + 19\;\vec{j}.

<h3>2.</h3>

\displaystyle \begin{aligned}8\;\vec{u} - 6\;\vec{v} &= 8\;\left(\begin{array}{c}11\\12\end{array}\right) + (-6) \;\left(\begin{array}{c}-16\\6\end{array}\right)\\&=\left(\begin{array}{c}88+96\\96 - 36\end{array}\right)\\&= \left(\begin{array}{c}184\\60\end{array}\right)= \langle 184, 60\rangle\\&=184\;\vec{i} + 60\;\vec{j} \end{aligned}.

<h3>3.</h3>

\displaystyle \begin{aligned}7\;\vec{v} - 4\;\vec{w} &= 7\;\left(\begin{array}{c}-16\\6\end{array}\right) + (-4) \;\left(\begin{array}{c}4\\-5\end{array}\right)\\&=\left(\begin{array}{c}-112 - 16\\42+20\end{array}\right)\\&= \left(\begin{array}{c}-128\\62\end{array}\right)= \langle -128, 62\rangle\\&=-128\;\vec{i} + 62\;\vec{j} \end{aligned}.

<h3>4.</h3>

\displaystyle \begin{aligned}\;\vec{u} - 5\;\vec{w} &= \left(\begin{array}{c}11\\12\end{array}\right) + (-5) \;\left(\begin{array}{c}4\\-5\end{array}\right)\\&=\left(\begin{array}{c}11-20\\12+25\end{array}\right)\\&= \left(\begin{array}{c}-9\\37\end{array}\right)= \langle -9, 37\rangle\\&=-9\;\vec{i} + 37\;\vec{j} \end{aligned}.

You might be interested in
How many ways can 4 people sit in a row of 7 ​chairs?
Ghella [55]

7\cdot6\cdot5\cdot4=840

6 0
3 years ago
Read 2 more answers
What additional information could be used to prove that △ABC ~ △NML? Check all that apply. ∠B ≅ ∠M △ABC is a right triangle. △AB
Elena L [17]

Answer:

1,3,5

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Find the values for a and b.
IrinaK [193]

Answer:

the value of a and b is 68

Step-by-step explanation:

the other half is 224 so you minus 360 from 224 which will give us 136 then you divide by 2 to get a and b.

6 0
3 years ago
Read 2 more answers
Jason sells computer software. He earns a base salary of $2,766.79 per month. In addition, he earns 9% of his total monthly sale
ipn [44]
Let's start with saying he earned 2766.79. He he also earns 9% of his sales.

9% of something is just .09 multiplied by that something. So if he earns 9% of 3743.37 plus the 2766.79, then we can say

y= 0.09 * 3743.37 + 2766.79
y = 3,103.69
5 0
3 years ago
Please Help! I got an answer, but I think it's wrong.
Ghella [55]

Answer:

I am pretty sure u got it right

Step-by-step explanation:

:)

3 0
2 years ago
Read 2 more answers
Other questions:
  • A video streaming service offers unlimited movies for $15 per month or $1.99 per movie. Write an inequality that represents when
    8·1 answer
  • The parabola below is a graph of the equation, -y+x=6. Which of the points satisfy the inequality, -y+x ≤ -6 ? Check all that ap
    8·1 answer
  • If b is between a and c find the value of x and the measure of line bc
    11·1 answer
  • The area of a playground is 320 yd2. The width of the playground is 4 yd longer than its length. Find the length and width of th
    8·2 answers
  • There are _____ min in two fifth of 2 hours
    8·1 answer
  • At a business meeting every person shakes each other's hand once if they were 91 handshakes in total the number of people at the
    5·1 answer
  • 18. Solve the quadratic equation by using the graph.<br><br> Look at screenshot
    6·1 answer
  • Find the sum: −11−7−3+1+⋯+225
    15·2 answers
  • Whoever answers correctly gets brainlist!
    6·1 answer
  • Find the Equation of the lines.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!