 A vector can be multiplied by another vector but may not be divided by another vector. There are two kinds of products of vectors used broadly in physics and engineering. One kind of multiplication is a scalar multiplication of two vectors. Taking a scalar product of two vectors results in a number (a scalar), as its name indicates. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector. A quite different kind of multiplication is a vector multiplication of vectors. Taking a vector product of two vectors returns as a result a vector, as its name suggests. Vector products are used to define other derived vector quantities. For example, in describing rotations, a vector quantity called torque is defined as a vector product of an applied force (a vector) and its distance from pivot to force (a vector). It is important to distinguish between these two kinds of vector multiplications because the scalar product is a scalar quantity and a vector product is a vector quantity.
Scalar multiplication of two vectors yields a scalar product.
The scalar product
of two vectors
and
is a number defined by the equation
where
is the angle between the vectors (shown in (Figure)). The scalar product is also called the dot product because of the dot notation that indicates it. In the definition of the dot product, the direction of angle
does not matter, and
can be measured from either of the two vectors to the other because
. The dot product is a negative number when
and is a positive number when
. Moreover, the dot product of two parallel vectors is
, and the dot product of two antiparallel vectors is
. The scalar product of two orthogonal vectors vanishes:
. The scalar product of a vector with itself is the square of its magnitude:
of vector
onto the direction of vector
. (c) The orthogonal projection
of vector
onto the direction of vector
.
For the vectors shown in (Figure), find the scalar product
. StrategyFrom (Figure), the magnitudes of vectors
and
are A = 10.0 and F = 20.0. Angle , between them, is the difference:
. Substituting these values into (Figure) gives the scalar product. Solution[reveal-answer q=”447394″]Show Answer[/reveal-answer]
[/hidden-answer]
For the vectors given in (Figure), find the scalar products
and
.
[reveal-answer q=”fs-id1167131172251″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131172251″]
,
[/hidden-answer] In the Cartesian coordinate system, scalar products of the unit vector of an axis with other unit vectors of axes always vanish because these unit vectors are orthogonal:
In these equations, we use the fact that the magnitudes of all unit vectors are one:
. For unit vectors of the axes, (Figure) gives the following identities:
The scalar product
can also be interpreted as either the product of B with the orthogonal projection
of vector
onto the direction of vector
((Figure)(b)) or the product of A with the orthogonal projection
of vector
onto the direction of vector
((Figure)(c)):
For example, in the rectangular coordinate system in a plane, the scalar x-component of a vector is its dot product with the unit vector , and the scalar y-component of a vector is its dot product with the unit vector :
Scalar multiplication of vectors is commutative,
and obeys the distributive law:
We can use the commutative and distributive laws to derive various relations for vectors, such as expressing the dot product of two vectors in terms of their scalar components.
For vector
in a rectangular coordinate system, use (Figure) through (Figure) to show that
and
. When the vectors in (Figure) are given in their vector component forms,
we can compute their scalar product as follows:
Since scalar products of two different unit vectors of axes give zero, and scalar products of unit vectors with themselves give one (see (Figure) and (Figure)), there are only three nonzero terms in this expression. Thus, the scalar product simplifies to
We can use (Figure) for the scalar product in terms of scalar components of vectors to find the angle between two vectors. When we divide (Figure) by AB, we obtain the equation for
, into which we substitute (Figure):
Angle
between vectors
and
is obtained by taking the inverse cosine of the expression in (Figure).
Three dogs are pulling on a stick in different directions, as shown in (Figure). The first dog pulls with force
, the second dog pulls with force
, and the third dog pulls with force
. What is the angle between forces
and
?
The components of force vector
are
,
, and
, whereas those of force vector
are
,
, and
. Computing the scalar product of these vectors and their magnitudes, and substituting into (Figure) gives the angle of interest. Solution[reveal-answer q=”653304″]Show Answer[/reveal-answer]
and
are
and
Substituting the scalar components into (Figure) yields the scalar product
Finally, substituting everything into (Figure) gives the angle
[/hidden-answer] SignificanceNotice that when vectors are given in terms of the unit vectors of axes, we can find the angle between them without knowing the specifics about the geographic directions the unit vectors represent. Here, for example, the +x-direction might be to the east and the +y-direction might be to the north. But, the angle between the forces in the problem is the same if the +x-direction is to the west and the +y-direction is to the south.
Find the angle between forces
and
in (Figure).
[reveal-answer q=”fs-id1167131496552″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131496552″]
[/hidden-answer]
When force
pulls on an object and when it causes its displacement
, we say the force performs work. The amount of work the force does is the scalar product
. If the stick in (Figure) moves momentarily and gets displaced by vector
, how much work is done by the third dog in (Figure)? StrategyWe compute the scalar product of displacement vector
with force vector
, which is the pull from the third dog. Let’s use
to denote the work done by force
on displacement
. Solution[reveal-answer q=”560347″]Show Answer[/reveal-answer]
[/hidden-answer] SignificanceThe SI unit of work is called the joule
, where 1 J = 1
. The unit
can be written as
, so the answer can be expressed as
.
How much work is done by the first dog and by the second dog in (Figure) on the displacement in (Figure)?
[reveal-answer q=”fs-id1167130006416″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167130006416″]
,
[/hidden-answer]
Vector multiplication of two vectors yields a vector product.
The vector product of two vectors
and
is denoted by
and is often referred to as a cross product. The vector product is a vector that has its direction perpendicular to both vectors
and
. In other words, vector
is perpendicular to the plane that contains vectors
and
, as shown in (Figure). The magnitude of the vector product is defined as
where angle
, between the two vectors, is measured from vector
(first vector in the product) to vector
(second vector in the product), as indicated in (Figure), and is between and
. According to (Figure), the vector product vanishes for pairs of vectors that are either parallel
or antiparallel
because
.
is a vector perpendicular to the plane that contains vectors
and
. Small squares drawn in perspective mark right angles between
and
, and between
and
so that if
and
lie on the floor, vector
points vertically upward to the ceiling. (b) The vector product
is a vector antiparallel to vector
. On the line perpendicular to the plane that contains vectors
and
there are two alternative directions—either up or down, as shown in (Figure)—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
points upward, as seen in (Figure)(a). If we reverse the order of multiplication, so that now
comes first in the product, then vector
must point downward, as seen in (Figure)(b). This means that vectors
and
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:
The corkscrew right-hand rule is a common mnemonic used to determine the direction of the vector product. As shown in (Figure), a corkscrew is placed in a direction perpendicular to the plane that contains vectors
and
, and its handle is turned in the direction from the first to the second vector in the product. The direction of the cross product is given by the progression of the corkscrew.
. Place a corkscrew in the direction perpendicular to the plane that contains vectors
and
, and turn it in the direction from the first to the second vector in the product. The direction of the cross product is given by the progression of the corkscrew. (a) Upward movement means the cross-product vector points up. (b) Downward movement means the cross-product vector points downward.
The mechanical advantage that a familiar tool called a wrench provides ((Figure)) depends on magnitude F of the applied force, on its direction with respect to the wrench handle, and on how far from the nut this force is applied. The distance R from the nut to the point where force vector
is attached and is represented by the radial vector
. The physical vector quantity that makes the nut turn is called torque (denoted by
, and it is the vector product of the distance between the pivot to force with the force:
. To loosen a rusty nut, a 20.00-N force is applied to the wrench handle at angle
and at a distance of 0.25 m from the nut, as shown in (Figure)(a). Find the magnitude and direction of the torque applied to the nut. What would the magnitude and direction of the torque be if the force were applied at angle
, as shown in (Figure)(b)? For what value of angle
does the torque have the largest magnitude?
We adopt the frame of reference shown in (Figure), where vectors
and
lie in the xy-plane and the origin is at the position of the nut. The radial direction along vector
(pointing away from the origin) is the reference direction for measuring the angle
because
is the first vector in the vector product
. Vector
must lie along the z-axis because this is the axis that is perpendicular to the xy-plane, where both
and
lie. To compute the magnitude , we use (Figure). To find the direction of
, we use the corkscrew right-hand rule ((Figure)). Solution[reveal-answer q=”915163″]Show Answer[/reveal-answer]
in the positive direction of the z-axis. Physically, it means the torque vector
points out of the page, perpendicular to the wrench handle. We identify F = 20.00 N and R = 0.25 m, and compute the magnitude using (Figure):
For the situation in (b), the corkscrew rule gives the direction of
in the negative direction of the z-axis. Physically, it means the vector
points into the page, perpendicular to the wrench handle. The magnitude of this torque is
The torque has the largest value when
, which happens when
. Physically, it means the wrench is most effective—giving us the best mechanical advantage—when we apply the force perpendicular to the wrench handle. For the situation in this example, this best-torque value is
.[/hidden-answer] SignificanceWhen solving mechanics problems, we often do not need to use the corkscrew rule at all, as we’ll see now in the following equivalent solution. Notice that once we have identified that vector
lies along the z-axis, we can write this vector in terms of the unit vector of the z-axis:
In this equation, the number that multiplies is the scalar z-component of the vector
. In the computation of this component, care must be taken that the angle
is measured counterclockwise from
(first vector) to
(second vector). Following this principle for the angles, we obtain
for the situation in (a), and we obtain
for the situation in (b). In the latter case, the angle is negative because the graph in (Figure) indicates the angle is measured clockwise; but, the same result is obtained when this angle is measured counterclockwise because
and
. In this way, we obtain the solution without reference to the corkscrew rule. For the situation in (a), the solution is
; for the situation in (b), the solution is
.
For the vectors given in (Figure), find the vector products
and
.
[reveal-answer q=”fs-id1167131635182″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131635182″]
or, equivalently,
, and the direction is into the page;
or, equivalently,
, and the direction is out of the page. Similar to the dot product ((Figure)), the cross product has the following distributive property:
The distributive property is applied frequently when vectors are expressed in their component forms, in terms of unit vectors of Cartesian axes. When we apply the definition of the cross product, (Figure), to unit vectors , , and that define the positive x-, y-, and z-directions in space, we find that
All other cross products of these three unit vectors must be vectors of unit magnitudes because , , and are orthogonal. For example, for the pair and , the magnitude is
. The direction of the vector product
must be orthogonal to the xy-plane, which means it must be along the z-axis. The only unit vectors along the z-axis are or
. By the corkscrew rule, the direction of vector
must be parallel to the positive z-axis. Therefore, the result of the multiplication
is identical to
. We can repeat similar reasoning for the remaining pairs of unit vectors. The results of these multiplications are
Notice that in (Figure), the three unit vectors , , and appear in the cyclic order shown in a diagram in (Figure)(a). The cyclic order means that in the product formula, follows and comes before , or follows and comes before , or follows and comes before . The cross product of two different unit vectors is always a third unit vector. When two unit vectors in the cross product appear in the cyclic order, the result of such a multiplication is the remaining unit vector, as illustrated in (Figure)(b). When unit vectors in the cross product appear in a different order, the result is a unit vector that is antiparallel to the remaining unit vector (i.e., the result is with the minus sign, as shown by the examples in (Figure)(c) and (Figure)(d). In practice, when the task is to find cross products of vectors that are given in vector component form, this rule for the cross-multiplication of unit vectors is very useful.
Suppose we want to find the cross product
for vectors
and
. We can use the distributive property ((Figure)), the anticommutative property ((Figure)), and the results in (Figure) and (Figure) for unit vectors to perform the following algebra:
When performing algebraic operations involving the cross product, be very careful about keeping the correct order of multiplication because the cross product is anticommutative. The last two steps that we still have to do to complete our task are, first, grouping the terms that contain a common unit vector and, second, factoring. In this way we obtain the following very useful expression for the computation of the cross product:
In this expression, the scalar components of the cross-product vector are
When finding the cross product, in practice, we can use either (Figure) or (Figure), depending on which one of them seems to be less complex computationally. They both lead to the same final result. One way to make sure if the final result is correct is to use them both.
When moving in a magnetic field, some particles may experience a magnetic force. Without going into details—a detailed study of magnetic phenomena comes in later chapters—let’s acknowledge that the magnetic field
is a vector, the magnetic force
is a vector, and the velocity
of the particle is a vector. The magnetic force vector is proportional to the vector product of the velocity vector with the magnetic field vector, which we express as
. In this equation, a constant takes care of the consistency in physical units, so we can omit physical units on vectors
and
. In this example, let’s assume the constant is positive. A particle moving in space with velocity vector
enters a region with a magnetic field and experiences a magnetic force. Find the magnetic force
on this particle at the entry point to the region where the magnetic field vector is (a)
and (b)
. In each case, find magnitude F of the magnetic force and angle the force vector
makes with the given magnetic field vector
. StrategyFirst, we want to find the vector product
, because then we can determine the magnetic force using
. Magnitude F can be found either by using components,
, or by computing the magnitude
directly using (Figure). In the latter approach, we would have to find the angle between vectors
and
. When we have
, the general method for finding the direction angle involves the computation of the scalar product
and substitution into (Figure). To compute the vector product we can either use (Figure) or compute the product directly, whichever way is simpler. Solution[reveal-answer q=”230259″]Show Answer[/reveal-answer]
,
, and
.
,
, and
. Substituting them into (Figure) gives the scalar components of vector
:
Thus, the magnetic force is
and its magnitude is
To compute angle , we may need to find the magnitude of the magnetic field vector,
and the scalar product
:
Now, substituting into (Figure) gives angle :
Hence, the magnetic force vector is perpendicular to the magnetic field vector. (We could have saved some time if we had computed the scalar product earlier.) (b) Because vector
has only one component, we can perform the algebra quickly and find the vector product directly:
The magnitude of the magnetic force is
Because the scalar product is
the magnetic force vector
is perpendicular to the magnetic field vector
.[/hidden-answer] SignificanceEven without actually computing the scalar product, we can predict that the magnetic force vector must always be perpendicular to the magnetic field vector because of the way this vector is constructed. Namely, the magnetic force vector is the vector product
and, by the definition of the vector product (see (Figure)), vector
must be perpendicular to both vectors
and
.
Given two vectors
and
, find (a)
, (b)
, (c) the angle between
and
, and (d) the angle between
and vector
.
[reveal-answer q=”fs-id1167134946260″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167134946260″] a.
, b. 2, c.
, d.
[/hidden-answer] In conclusion to this section, we want to stress that “dot product” and “cross product” are entirely different mathematical objects that have different meanings. The dot product is a scalar; the cross product is a vector. Later chapters use the terms dot product and scalar product interchangeably. Similarly, the terms cross product and vector product are used interchangeably.
Assuming the +x-axis is horizontal to the right for the vectors in the following figure, find the following scalar products: (a)
, (b)
, (c)
, (d)
, (e)
, (f)
, (g)
, and (h)
.
Assuming the +x-axis is horizontal to the right for the vectors in the preceding figure, find (a) the component of vector
along vector
, (b) the component of vector
along vector
, (c) the component of vector along vector
, and (d) the component of vector
along vector .
[reveal-answer q=”fs-id1167131466722″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131466722″] a. 8.66, b. 10.39, c. 0.866, d. 17.32 [/hidden-answer]
Find the angle between vectors for (a)
and
and (b)
and
.
Find the angles that vector
makes with the x-, y-, and z– axes.
[reveal-answer q=”fs-id1167130204653″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167130204653″]
[/hidden-answer]
Show that the force vector
is orthogonal to the force vector
.
Assuming the +x-axis is horizontal to the right for the vectors in the previous figure, find the following vector products: (a)
, (b)
, (c)
, (d)
, (e)
, (f)
, (g)
, and (h)
.
[reveal-answer q=”fs-id1167131591244″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131591244″] a.
, b.
, c.
, d.
, e.
, f.
, g.
, h. 0
Find the cross product
for (a)
and
, (b)
and
, (c)
and
, and (d)
and
.
For the vectors in the earlier figure, find (a)
, (b)
, and (c)
.
[reveal-answer q=”fs-id1167131575250″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131575250″] a. 0, b. 173,194, c.
[/hidden-answer]
(a) If
, can we conclude
? (b) If
, can we conclude
? (c) If
, can we conclude
? Why or why not?
You fly
in a straight line in still air in the direction
south of west. (a) Find the distances you would have to fly due south and then due west to arrive at the same point. (b) Find the distances you would have to fly first in a direction
south of west and then in a direction
west of north. Note these are the components of the displacement along a different set of axes—namely, the one rotated by
with respect to the axes in (a).
[reveal-answer q=”fs-id1167131608214″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131608214″] a. 18.4 km and 26.2 km, b. 31.5 km and 5.56 km [/hidden-answer]
Rectangular coordinates of a point are given by (2, y) and its polar coordinates are given by
. Find y and r.
If the polar coordinates of a point are
and its rectangular coordinates are
, determine the polar coordinates of the following points: (a) (−x, y), (b) (−2x, −2y), and (c) (3x, −3y).
[reveal-answer q=”fs-id1167134887251″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167134887251″] a.
, b.
, (c)
[/hidden-answer]
Vectors
and
have identical magnitudes of 5.0 units. Find the angle between them if
.
Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops at the islands of Noi and Poi. It sails from Moi for 4.76 nautical miles (nmi) in a direction
north of east to Noi. From Noi, it sails
west of north to Poi. On its return leg from Poi, it sails
east of south. What distance does the boat sail between Noi and Poi? What distance does it sail between Moi and Poi? Express your answer both in nautical miles and in kilometers. Note: 1 nmi = 1852 m.
[reveal-answer q=”fs-id1167131436626″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131436626″]
[/hidden-answer]
An air traffic controller notices two signals from two planes on the radar monitor. One plane is at altitude 800 m and in a 19.2-km horizontal distance to the tower in a direction
south of west. The second plane is at altitude 1100 m and its horizontal distance is 17.6 km and
south of west. What is the distance between these planes?
Show that when
, then
, where
is the angle between vectors
and
.
[reveal-answer q=”fs-id1167131501386″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131501386″] proof [/hidden-answer]
Four force vectors each have the same magnitude f. What is the largest magnitude the resultant force vector may have when these forces are added? What is the smallest magnitude of the resultant? Make a graph of both situations.
A skater glides along a circular path of radius 5.00 m in clockwise direction. When he coasts around one-half of the circle, starting from the west point, find (a) the magnitude of his displacement vector and (b) how far he actually skated. (c) What is the magnitude of his displacement vector when he skates all the way around the circle and comes back to the west point?
[reveal-answer q=”fs-id1167131128637″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131128637″] a. 10.00 m, b.
, c. 0
A stubborn dog is being walked on a leash by its owner. At one point, the dog encounters an interesting scent at some spot on the ground and wants to explore it in detail, but the owner gets impatient and pulls on the leash with force
along the leash. (a) What is the magnitude of the pulling force? (b) What angle does the leash make with the vertical?
If the velocity vector of a polar bear is
, how fast and in what geographic direction is it heading? Here, and are directions to geographic east and north, respectively.
[reveal-answer q=”fs-id1167130010413″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167130010413″] 22.2 km/h,
south of west
Find the scalar components of three-dimensional vectors
and
in the following figure and write the vectors in vector component form in terms of the unit vectors of the axes.
A diver explores a shallow reef off the coast of Belize. She initially swims 90.0 m north, makes a turn to the east and continues for 200.0 m, then follows a big grouper for 80.0 m in the direction
north of east. In the meantime, a local current displaces her by 150.0 m south. Assuming the current is no longer present, in what direction and how far should she now swim to come back to the point where she started?
[reveal-answer q=”fs-id1167130051819″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167130051819″] 240.2 m,
south of west
A force vector
has x– and y-components, respectively, of −8.80 units of force and 15.00 units of force. The x– and y-components of force vector
are, respectively, 13.20 units of force and −6.60 units of force. Find the components of force vector
that satisfies the vector equation
.
Vectors
and
are two orthogonal vectors in the xy-plane and they have identical magnitudes. If
, find
.
[reveal-answer q=”fs-id1167131091310″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1167131091310″]
or
[/hidden-answer]
For the three-dimensional vectors in the following figure, find (a)
, (b)
, and (c)
.
Show that
is the volume of the parallelepiped, with edges formed by the three vectors in the following figure. |
If the two vectors A and b are antiparallel to each other then the their scalar product is given by
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