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
Amiraneli [1.4K]
3 years ago
11

The vector (a) is a multiple of the vector (2i +3j) and (b) is a multiple of (2i+5j) The sum (a+b) is a multiple of the vector (

8i +15j). Given that /a+b/= 34 and the scaler multiple of (8i+15j) is positive, Find the magnitude of a and b.​
Mathematics
1 answer:
kow [346]3 years ago
6 0

Answer:

\|a\| = 5\sqrt{13}.

\|b\| = 3\sqrt{29}.

Step-by-step explanation:

Let m,n, and k be scalars such that:

\displaystyle a = m\, (2\, \vec{i} + 3\, \vec{j}) = m\, \begin{bmatrix}2 \\ 3\end{bmatrix}.

\displaystyle b = n\, (2\, \vec{i} + 5\, \vec{j}) = n\, \begin{bmatrix}2 \\ 5\end{bmatrix}.

\displaystyle (a + b) = k\, (8\, \vec{i} + 15\, \vec{j}) = k\, \begin{bmatrix}8 \\ 15\end{bmatrix}.

The question states that \| a + b \| = 34. In other words:

k\, \sqrt{8^{2} + 15^{2}} = 34.

k^{2} \, (8^{2} + 15^{2}) = 34^{2}.

289\, k^{2} = 34^{2}.

Make use of the fact that 289 = 17^{2} whereas 34 = 2 \times 17.

\begin{aligned}17^{2}\, k^{2} &= 34^{2}\\ &= (2 \times 17)^{2} \\ &= 2^{2} \times 17^{2} \end{aligned}.

k^{2} = 2^{2}.

The question also states that the scalar multiple here is positive. Hence, k = 2.

Therefore:

\begin{aligned} (a + b) &= k\, (8\, \vec{i} + 15\, \vec{j}) \\ &= 2\, (8\, \vec{i} + 15\, \vec{j}) \\ &= 16\, \vec{i} + 30\, \vec{j}\\ &= \begin{bmatrix}16 \\ 30 \end{bmatrix}\end{aligned}.

(a + b) could also be expressed in terms of m and n:

\begin{aligned} a + b &= m\, (2\, \vec{i} + 3\, \vec{j}) + n\, (2\, \vec{i} + 5\, \vec{j}) \\ &= (2\, m + 2\, n) \, \vec{i} + (3\, m + 5\, n) \, \vec{j} \end{aligned}.

\begin{aligned} a + b &= m\, \begin{bmatrix}2\\ 3 \end{bmatrix} + n\, \begin{bmatrix} 2\\ 5 \end{bmatrix} \\ &= \begin{bmatrix}2\, m + 2\, n \\ 3\, m + 5\, n\end{bmatrix}\end{aligned}.

Equate the two expressions and solve for m and n:

\begin{cases}2\, m + 2\, n = 16 \\ 3\, m + 5\, n = 30\end{cases}.

\begin{cases}m = 5 \\ n = 3\end{cases}.

Hence:

\begin{aligned} \| a \| &= \| m\, (2\, \vec{i} + 3\, \vec{j})\| \\ &= m\, \| (2\, \vec{i} + 3\, \vec{j}) \| \\ &= 5\, \sqrt{2^{2} + 3^{2}} = 5 \sqrt{13}\end{aligned}.

\begin{aligned} \| b \| &= \| n\, (2\, \vec{i} + 5\, \vec{j})\| \\ &= n\, \| (2\, \vec{i} + 5\, \vec{j}) \| \\ &= 3\, \sqrt{2^{2} + 5^{2}} = 3 \sqrt{29}\end{aligned}.

You might be interested in
Please help!! Is this statement always,sometimes,or never true
Kipish [7]
Always because the higher the negative he less its and the same for he positive side 
4 0
3 years ago
Read 2 more answers
Write an equation, in slope-intercept form, of the line that passes through the given point and satisfies the given condition. (
malfutka [58]

Answer:

y = -1/2x - 9/2

Step-by-step explanation:

8 0
3 years ago
Bags of sugar come in 3 sizes: Small bag: A 250 g bag costs 95p. Medium bag: A 500 g bag costs £1.80. Large bag: A 3 kg bag cost
azamat

Answer:

see below

Step-by-step explanation:

Small bag

95p / 250 g * 12/12 to get to 3 kg = 1140 p / 3000g =11.40£s/ 3 kgs

Medium bag

1.8£  / 500 g * 6/6 to get to 3 kg = 10.8£ / 3000g =10.80£/ 3 kgs

Large bag

3.7£  / 3 kg *

6 0
3 years ago
The ratio of blue paint to yellow paint in a green paint mixture is 2:3. If you use 16 ounces of blue paint how much ounces of y
Verizon [17]

Answer:

Step-by-step explanation:

2 in 2:3 is the amount of blue paint

3 in 2:3 is the amount of yellow paint

Figure out how many times to multiply 2 to get 16:

16/2=8

Now, multiply 3 by 8, which would be equal to 24.

Ratio: 16:24

8 0
2 years ago
Find the surface area of each prism
goldfiish [28.3K]

Answer:

Step-by-step explanation:

2.

Surface area = ( 6 *10 ) *3  + ( ( 7*4.87) /2)*2 = 180 + 34.09 = 214.09

3.

Surface area = (5*10) *3 + (( 7*4.2)/2)*2 = 150 +29.4 = 179.4

6 0
3 years ago
Other questions:
  • Prove that there are infinitely many primes of the form 4k + 3, where k is a non-negative integer. [Hint: Suppose that there are
    14·2 answers
  • Horatio is purchasing a computer cable for $15.49.If the sales tax rate in his state is 5.25% what is the total cost of the purc
    10·1 answer
  • Triple A number is the same as double the number increased by 5
    5·2 answers
  • 24 POINTS IF ANSWERED CORRECTLY.
    8·2 answers
  • a system of linear equations include the line that is created by the equation y=0.5x-1 and the linethrough the points (3, 1) and
    15·2 answers
  • 1.7, 8/5, 2% ascending
    9·1 answer
  • By the distributive law 49x17 + 49x3 = ?
    15·1 answer
  • Choose the graph of the following equation 64(x+1)^2+36y^2=48^2
    10·1 answer
  • Line a: 4y + x = 8<br> Line b: 2y + r = 4<br> Line c: 22 - 3x + 6
    8·1 answer
  • Please help this is due today!!
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!