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:

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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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