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
lapo4ka [179]
3 years ago
8

onsider the vectors a→=3i→+j→−k→,b→=−3i→−j→+k→,c→=i→+0.333333j→+0.333333k→ d→=i→+3j→+4k→,g→=i→+3j→−k→. Which pairs (if any) of t

hese vectors are (a) Are perpendicular? (Enter none or a pair or list of pairs, e.g., if a→ is perpendicular to b→ and c→, enter (a,b),(a,c).) (b) Are parallel? (Enter none or a pair or list of pairs, e.g., if a→ is parallel to b→ and c→, enter (a,b),(a,c).) (c) Have an angles less than π/2 between them? (Enter none or a pair or list of pairs, e.g., if a→ is at an angle less than pi/2 from b→ and c→, enter (a,b),(a,c).) (d) Have an angle of more than π/2 between them? (Enter none or a pair or list of pairs, e.g., if a→ is at an angle greater than pi/2 from b→ and c→, enter (a,b),(a,c).)
Mathematics
1 answer:
Fed [463]3 years ago
5 0

We're given the vectors

\vec a = 3\vec\imath + \vec\jmath - \vec k = \langle3,1,-1\rangle \\\\ \vec b = -3\,\vec\imath-\vec\jmath + \vec k = \langle-3,-1,1\rangle \\\\ \vec c = \vec\imath + \dfrac13\,\vec\jmath + \dfrac13\,\vec k = \left\langle1,\dfrac13,\dfrac13\right\rangle \\\\ \vec d = \vec\imath + 3\,\vec\jmath + 4\,\vec k = \langle 1,3,4\rangle \\\\ \vec g = \vec\imath + 3\,\vec\jmath - \vec k = \langle1,3,-1\rangle

(a) Two vectors are perpendicular if their dot product is zero. For instance, \vec a and \vec b are not perpendicular because

\vec a\cdot\vec b = \langle3,1,-1\rangle\cdot\langle-3,-1,1\rangle = 3\times(-3)+1\times(-1)+(-1)\times1 = -11

You'll find that none of these vectors taken two at a time are perpendicular to each other.

(b) Recall for any two vectors \vec x and \vec y that

\vec x\cdot\vec y = \|\vec x\| \|\vec y\| \cos(\theta)

where \theta is the angle between \vec x and \vec y. If these vectors are parallel, then the angle between them is 0 rad or π rad, meaning they point in the same or in opposite directions, respectively.

We have cos(0) = 1 and cos(π) = -1, so

\vec x\cdot\vec y = \pm\|\vec x\| \|\vec y\|

For instance, we know that

\vec a\cdot\vec b = -11

and we have

\|\vec a\| = \sqrt{3^2 + 1^2 + (-1)^2} = \sqrt{11} \\\\ \|\vec b\| = \sqrt{(-3)^2+(-1)^2+1^2} = \sqrt{11}

so \vec a and \vec b are indeed parallel and point in opposite directions, since -11 = - √11 × √11.

On the other hand, \vec a and \vec c are not parallel, since

\vec a\cdot\vec c = \langle3,1,-1\rangle\cdot\left\langle1,\dfrac13,\dfrac13\right\rangle = 3\times1+1\times\dfrac13+(-1)\times\dfrac13 = 3 \\\\ \|\vec a\|\|\vec c\| = \sqrt{3^2+1^2+(-1)^2}\times\sqrt{1^2+\dfrac1{3^2}+\dfrac1{3^2}} = \dfrac{11}3

and clearly 3 ≠ ±11/3.

It turns out that (a, b) is the only pair of parallel vectors.

(c) The cosine of an angle measuring between 0 and π/2 rad is positive, so you just need to check the sign of

\cos(\theta) = \dfrac{\vec x\cdot\vec y}{\|\vec x\|\|\vec y\|}

For instance, we know \vec a and \vec b are parallel and have an angle of π rad between them. cos(π) = -1, so this pair doesn't qualify. Meanwhile, the angle between

\cos(\theta)=\dfrac3{\frac{11}3}\right) =\dfrac9{11} > 0

so \vec a and \vec c do qualify.

You'd find that the pairs ((a, c), (a, d), (a, g), (c, d), (c, g), (d, g)).

(d) An angle between π/2 and π has a negative cosine. None of the vectors are perpendicular to each other, so this happens for the remaining pairs, ((a, b), (b, c), (b, d), (b, g)).

You might be interested in
Can someone please help me with this, find the unit rate cost per item which is these chips set up the equation and show the wor
user100 [1]

Answer:$3

Step-by-step explanation:

You always divide y by x. In this case 6/2 when you set up a chart. When you do this you get $3 unit rate.

1 (unit rate)             2

___________=_____

x (unit rate price)   6

Use cross products

1(6)=x(2)

6=2x

_____

2     2

x=3

4 0
3 years ago
Read 2 more answers
Is the following statement TRUE or FALSE? Select the best choice from the following to justify your
ale4655 [162]

Answer:

first \: answer \: is \: true

5 0
3 years ago
1. Tim goes to the supermarket and buys
Luba_88 [7]

Answer: 12

Step-by-step explanation:  because each pack is 50cents so multiply .50 by 24 so the answer is 12

3 0
2 years ago
What is 235 divided by 5?
Alex Ar [27]
The answer to this question is: 47
4 0
3 years ago
Read 2 more answers
Pleaseee answer quick
FinnZ [79.3K]

Answer: m

Step-by-step explanation:

if you are talking about a specific point, then it would be m because OA, AB and

<-->

AB

all have other points to them.

AB is when you are trying to explain the distance between AB

m would be used just to figure out where that point on the grid really is

6 0
2 years ago
Other questions:
  • PLEASE HELP I DONT UNDERSTAND
    15·1 answer
  • How far will you travel if you run for 10 minutes at 2m/sec
    12·1 answer
  • Blake and his children went into a bakery and where they sell donuts for $1.50 each and cookies for $0.50 each. Blake has $15 to
    6·1 answer
  • Which expressions are equivalent to 2^5.2^4? Check all that apply.
    10·2 answers
  • The slope of a line ST = 1. What is the value of y if S = (−5,−3) and T = (−1,y).
    8·1 answer
  • 3× - 2y = 13, 2x +2y =0​
    7·1 answer
  • A group that was given immigration priority under the Immigration Act of 1965
    10·1 answer
  • Solve the question 35&gt;-5(4-x)
    14·1 answer
  • Find the HCF of the following by<br>Pime factorisation.<br>a) 72,54,27<br>b) 24, 16, 36​
    13·2 answers
  • Evaluate the expression for x = 2 and y = - 7.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!