Vectors which of the following can be computed

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.

Three dogs are pulling on a stick in different directions, as shown in Figure. Computing the scalar product of these vectors and their magnitudes, and substituting into Figure gives the angle of interest.

Notice 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. How much work is done by the first dog and by the second dog in Figure on the displacement in Figure? The magnitude of the vector product is defined as. 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. The direction of the cross product is given by the progression of the corkscrew.

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. To loosen a rusty nut, a Find the magnitude and direction of the torque applied to the nut.

The magnitude of this torque is. Physically, it means the wrench is most effective—giving us the best mechanical advantage—when we apply the force perpendicular to the wrench handle. In this way, we obtain the solution without reference to the corkscrew rule. 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. We can repeat similar reasoning for the remaining pairs of unit vectors. The results of these multiplications are. 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. 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. These products have the positive sign. These products have the negative sign. 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. To compute the vector product we can either use Figure or compute the product directly, whichever way is simpler.

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. Even 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.

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. How can you correct them? We can use Equation 2. When we divide Equation 2. Substituting the scalar components into Equation 2.

Finally, substituting everything into Equation 2. How much work is done by the first dog and by the second dog in Example 2. The magnitude of the vector product is defined as. According to Equation 2. 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 2. The direction of the cross product is given by the progression of the corkscrew. To loosen a rusty nut, a Find the magnitude and direction of the torque applied to the nut.

The magnitude of this torque is. Physically, it means the wrench is most effective—giving us the best mechanical advantage—when we apply the force perpendicular to the wrench handle. In the latter case, the angle is negative because the graph in Figure 2. In this way, we obtain the solution without reference to the corkscrew rule. Similar to the dot product Equation 2. 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, Equation 2. We can repeat similar reasoning for the remaining pairs of unit vectors. The results of these multiplications are. Notice that in Equation 2. 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 2.

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. 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.

We can use the distributive property Equation 2. 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.

We define Definition Let u and v be a vectors. Then u can be broken up into two components, r and s such that r is parallel to v and s is perpendicular to v. The direction is correct since the right hand side of the formula is a constant multiple of v so the projection vector is in the direction of v as required. PQ Example Find the work done against gravity to move a 10 kg baby from the point 2,3 to the point 5,7?

Hence, it takes J of work to move the baby.