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
Reika [66]
3 years ago
5

Consider the points below. P(1, 0, 1), Q(−2, 1, 4), R(6, 2, 7) (a) Find a nonzero vector orthogonal to the plane through the poi

nts P, Q, and R. Correct: Your answer is correct. (b) Find the area of the triangle PQR.
Physics
1 answer:
kozerog [31]3 years ago
5 0

Answer:

a) (0, -33, 12)

b) area of the triangle : 17.55 units of area

Explanation:

<h2>a) </h2>

We know that the cross product of linearly independent vectors \vec{A} and \vec{B} gives us a nonzero, orthogonal to both, vector. So, if we can find two linearly independent vectors on the plane through the points P, Q, and R, we can use the cross product to obtain the answer to point a.

Luckily for us, we know that vectors \vec{A} = \vec{P}-\vec{Q} and \vec{B} = \vec{R} - \vec{Q} are living in the plane through the points P, Q, and R, and are linearly independent.

We know that they are linearly independent, cause to have one, and only one, plane through points P Q and R, this points must be linearly independent (as the dimension of a plane subspace is 3).

If they weren't linearly independent, we will obtain vector zero as the result of the cross product.

So, for our problem:

\vec{A} = \vec{P} - \vec{Q} \\\\\vec{A} = (1,0,1) - (-2,1,4)\\\\\vec{A} = (1 +2,0-1,1-4)\\\\\vec{A} = (3,-1,-3)

\vec{B} = \vec{R} - \vec{Q} \\\\\vec{B} = (6,2,7) - (-2,1,4)\\\\\vec{B} = (6 +2,2-1,7-4)\\\\\vec{B} = (8,1,3)

\vec{A} \times  \vec{B} = (A_y B_z - B_y A_z) \  \hat{i} - ( A_x B_z-B_xA_z) \ \hat{j} + (A_x B_y - B_x A_y ) \ \hat{k}

\vec{A} \times  \vec{B} = ( (-1) * 3 - 1 * (-3) ) \  \hat{i} - ( 3 * 3 - 8 * (-3)) \ \hat{j} + (3 * 1 - 8 * (-1) ) \ \hat{k}

\vec{A} \times  \vec{B} = ( - 3 + 3 ) \  \hat{i} - ( 9 + 24 ) \ \hat{j} + (3 + 8 ) \ \hat{k}

\vec{A} \times  \vec{B} = 0 \  \hat{i} - 33 \ \hat{j} + 12 \ \hat{k}

\vec{A} \times  \vec{B} =(0, -33, 12)

<h2>B)</h2>

We know that \vec{A} and \vec{B} are two sides of the triangle, and we also know that we can use the magnitude of the cross product to find the area of the triangle:

|\vec{A} \times  \vec{B} | = 2 * area_{triangle}

so:

\sqrt{(-33)^2 + (12)^2} = 2 * area_{triangle}

\sqrt{1233} = 2 * area_{triangle}

35.114= 2 * area_{triangle}

17.55 \ units \  of \ area =  area_{triangle}

You might be interested in
2.
saul85 [17]

Answer:

<u>B</u>

Explanation:

Planets have different year lengths because it depends how far they revolve from a celestial body. Each planet has its own orbital period. Planets closer to the star will have a lower orbital period compared to the ones that lie far away from it.

4 0
1 year ago
Read 2 more answers
A tennis ball is tossed upward with a speed of 3.0\,\dfrac{\text m}{\text s}3.0 s m ​ 3, point, 0, start fraction, start text, m
MrRa [10]

Answer:

The velocity of the ball is 0.92 m/s in the downward direction (-0.92 m/s).

Explanation:

Hi there!

The equation for the velocity of an object thrown upward is the following:

v = v0 + g · t

Where:

v = velocity of the ball.

v0 = initial velocity.

g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive).

t = time.

To find the velocity of the ball at t = 0.40 s, we have to replace "t" by 0.40 s in the equation:

v = v0 + g · t

v = 3.0 m/s - 9.8 m/s² · 0.40 s

v = -0.92 m/s

The velocity of the ball is 0.92 m/s in the downward direction (-0.92 m/s).

8 0
3 years ago
Read 2 more answers
Describe the full water cycle
sergey [27]

The water cycle (hydro-logic cycle), explains the constant motion of water on, above and below the surface of the Earth. The mass of water on Earth stays equally constant over time, although the severe portion of the water goes into the major reservoirs of ice, fresh water, saline water and atmospheric water is variable depending on a wide range of climatic variables. The water moves from one reservoir to another, such as from river to ocean, or from the ocean to the atmosphere, by the physical processes of evaporation, condensation, precipitation, infiltration, surface runoff, and subsurface flow. In doing so, the water goes through different forms: liquid, solid (ice) and vapor.

5 0
3 years ago
In what climates would you expect soil to form fastest? Why?
enyata [817]

Answer:

Soils develop faster in warm, moist climates and slowest in cold or arid ones. Rainfall is one of the most important climate factors in soil formation.

Please mark me brainliest!

Hope this helped God bless :)

5 0
3 years ago
A hot water heater is operated by using solar power. if the solar collector has an area of 5.3 m2 , and the power delivered by s
OLEGan [10]
Heat absorbed by the solar collector = Area*Irradiance = 5.3*995 = 5273.5 W

Heat Q in joules absorbed in t hours = Heat used to heat water. That is,

5273.5*t = mCΔT; where mass = volume*density = 1*1000 = 1000 kg

Therefore;
5273.5t = 1000*4186*(65-20) = 188370000
t = 188370000/5273.5 = 35720.11 seconds = 35720.11/(60*60) hours ≈ 9.92 hours.

It will take approximately 9.92 hours.
4 0
3 years ago
Other questions:
  • The 2-Mg truck is traveling at 15 m/s when the brakes on all its wheels are applied, causing it to skid for 10 m before coming t
    6·1 answer
  • A lithium atom has three protons, three neutrons, and three electrons. What is the overall charge on this atom?
    10·2 answers
  • A positively charged particle 1 is at the origin of a Cartesian coordinate system, and there are no other charged objects nearby
    13·1 answer
  • According to the Big Bang theory timeline, all matter and energy was compressed into a very small volume at an extremely high te
    9·1 answer
  • A student swings back and forth from position A to C, as shown. Which of the following happens when the swing moves from Positio
    13·2 answers
  • Why does water frozen in the cracks of a rock help to break down the rock? a) Water expands when frozen and chemically forces th
    9·1 answer
  • Compare and contrast an earthquake and a tsunami.
    15·1 answer
  • What type of electricity comes from an outlet​
    5·1 answer
  • At what speed should a ball of mass 2 kg be rolled in order to reach the other side of
    15·1 answer
  • How much is the velocity of a body when it travels 600m in 5 minutes?​
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!