2.4 Products of vectors (Page 3/16)

University physics volume 1 Page 3 / 16

Check Your Understanding Find the angle between forces ${\vec{F}}_{1}$ and ${\vec{F}}_{3}$ in [link] .

$131.9 °$

The work of a force

When force

\vec{F}

pulls on an object and when it causes its displacement

\vec{D}

, we say the force performs work. The amount of work the force does is the scalar product

\vec{F} \cdot \vec{D}

. If the stick in [link] moves momentarily and gets displaced by vector

\vec{D} = (−7.9 \hat{j} - 4.2 \hat{k}) cm

, how much work is done by the third dog in [link] ?

Strategy

We compute the scalar product of displacement vector

\vec{D}

with force vector

{\vec{F}}_{3} = (5.0 \hat{i} + 12.5 \hat{j}) N

, which is the pull from the third dog. Let’s use

W_{3}

to denote the work done by force

{\vec{F}}_{3}

on displacement

\vec{D}

Solution

Calculating the work is a straightforward application of the dot product:

\begin{matrix} W_{3} & = {\vec{F}}_{3} \cdot \vec{D} = F_{3 x} D_{x} + F_{3 y} D_{y} + F_{3 z} D_{z} \\ = (5.0 N) (0.0 cm) + (12.5 N) (−7.9 cm) + (0.0 N) (−4.2 cm) \\ = −98.7 N \cdot cm . \end{matrix}

Significance

The SI unit of work is called the joule

(J)

, where 1 J = 1

N \cdot m

. The unit

cm \cdot N

can be written as

10^{−2} m \cdot N = 10^{−2} J

, so the answer can be expressed as

W_{3} = −0.9875 J \approx −1.0 J

Got questions? Get instant answers now!

Check Your Understanding How much work is done by the first dog and by the second dog in [link] on the displacement in [link] ?

$W_{1} = 1.5 J$ , $W_{2} = 0.3 J$

Got questions? Get instant answers now!

The vector product of two vectors (the cross product)

Vector multiplication of two vectors yields a vector product.

Vector product (cross product)

The vector product of two vectors $\vec{A}$ and $\vec{B}$ is denoted by $\vec{A} \times \vec{B}$ and is often referred to as a cross product . The vector product is a vector that has its direction perpendicular to both vectors $\vec{A}$ and $\vec{B}$ . In other words, vector $\vec{A} \times \vec{B}$ is perpendicular to the plane that contains vectors $\vec{A}$ and $\vec{B}$ , as shown in [link] . The magnitude of the vector product is defined as

| \vec{A} \times \vec{B} | = A B sin φ,

where angle $φ$ , between the two vectors, is measured from vector $\vec{A}$ (first vector in the product) to vector $\vec{B}$ (second vector in the product), as indicated in [link] , and is between $0 °$ and $180 °$ .

According to [link] , the vector product vanishes for pairs of vectors that are either parallel $(φ = 0 °)$ or antiparallel $(φ = 180 °)$ because $sin 0 ° = sin 180 ° = 0$ .

Vector A points out and to the left, and vector B points out and to the right. The angle between them is phi. In figure a we are shown vector C which is the cross product of A cross B. Vector C points up and is perpendicular to both A and B. In figure b we are shown vector minus C which is the cross product of B cross A. Vector minus C points down and is perpendicular to both A and B. — The vector product of two vectors is drawn in three-dimensional space. (a) The vector product $\vec{A} \times \vec{B}$ is a vector perpendicular to the plane that contains vectors $\vec{A}$ and $\vec{B}$ . Small squares drawn in perspective mark right angles between $\vec{A}$ and $\vec{C}$ , and between $\vec{B}$ and $\vec{C}$ so that if $\vec{A}$ and $\vec{B}$ lie on the floor, vector $\vec{C}$ points vertically upward to the ceiling. (b) The vector product $\vec{B} \times \vec{A}$ is a vector antiparallel to vector $\vec{A} \times \vec{B}$ .

On the line perpendicular to the plane that contains vectors $\vec{A}$ and $\vec{B}$ there are two alternative directions—either up or down, as shown in [link] —and the direction of the vector product may be either one of them. In the standard right-handed orientation, where the angle between vectors is measured counterclockwise from the first vector, vector $\vec{A} \times \vec{B}$ points upward , as seen in [link] (a). If we reverse the order of multiplication, so that now $\vec{B}$ comes first in the product, then vector $\vec{B} \times \vec{A}$ must point downward , as seen in [link] (b). This means that vectors $\vec{A} \times \vec{B}$ and $\vec{B} \times \vec{A}$ are antiparallel to each other and that vector multiplication is not commutative but anticommutative . The anticommutative property means the vector product reverses the sign when the order of multiplication is reversed:

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

tijani

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John Reply

what is physics

Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

what is inorganic

emma Reply

what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

please, I'm a physics student and I need help in physics

Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

Ryan

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Maurice Reply

what are the types of wave

Maurice

answer

Magreth

progressive wave

Magreth

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Mohammed

Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo Reply

Who can show me the full solution in this problem?

Reofrir Reply

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Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

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