Outside Diameter of Conduit: Size: EMT: IMC: Rigid: Decimal: Fraction: Decimal: Fraction: Decimal: Fraction: 1/2' 0.706: 11/16: 0.815: 13/16: 0.84: 13/16: 3/4' 0.922. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.
Definition We define thedot product of two vectors v = ai + bjand w = ci + dj to be
v. w = ac + bd
Notice that the dot product of two vectors is a number and not a vector. For 3 dimensional vectors, we define the dot product similarly:
Definition of Direction Cosines Let v = ai + bj + ck be a vector, then we define the direction cosines to be the following:
a cos a = ||v||
b cos b = ||v||
c cos g = ||v||
Projections and Components Suppose that a car is stopped on a steep hill, and let g be the force of gravity acting on it. We can split the vector g into the component that is pushing the car down the road and the component that is pushing the car onto the road. We define
Definition
Let u and v be a vectors. Then ucan be broken up into two components, r and s such that r is parallel to v and s is perpendicular to v. ris called the projection ofuontov and s is called the component of u perpendicular to v.
We see that ||u|| ||v|| ||projvu || u.v = ||u|| ||v|| cos q = ||u|| = ||v|| ||projvu ||
hence
u . v ||projvu || = ||v||
We can calculate the projection of u onto v by the formula:
u . v projvu = v ||v||2
Notice that this works since if we take magnitudes of both sides we get that
u . v | |projvu|| = ||v|| ||v||2
and the right hand side simplifies to the formula above. 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.
Etrecheck 4 3 4 download free. To find the vector s, notice from the diagram that
projvu + s = u
so that
s = u - projv u The work done by a constant force F along PQ is given by
W = F.PQ
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Example
Find the work done against gravity to move a 10 kg baby from the point (2,3) to the point (5,7)?
Notice the negative sign verifies that the work is done against gravity. Hence, it takes 392 J of work to move the baby.
Suppose you are skiing and have a terrible fall. Your body spins around and you ski stays in place (do not try this at home). With proper bindings your bindings will release and your ski will come off. The bindings recognize that a force has been applied. This force is called torque. To compute it we use the cross produce of two vectors which not only gives the torque, but also produces the direction that is perpendicular to both the force and the direction of the leg.
The Cross Product Between Two Vectors
Definition Let u = ai + bj + ck and v = di + ej + fkbe vectors then we define the cross productv x wby the determinant of the matrix:
If you need more help see the lecture notes for Math 103 B on matrices.
Exercises Find u x v when
u = 3i + j - 2k, v = i - k
u = 2i - 4j - k, v = 3i - j + 2k
Notice that since switching the order of two rows of a determinant changes the sign of the determinant, we have u x v = -v x u
Geometry and the Cross Product
Let u and vbe vectors and consider the parallelogram that the two vectors make. Then ||u x v|| = Area of the Parallelogram
and the direction of u x v is a right angle to the parallelogram that follows the right hand rule
Note:Fori x j the magnitude is 1 and the direction isk, hence i x j = k.
Exercise Find j x k and i x k
Torque Revisited We define the torque (or the moment M of a force F about a point Q) as
M = PQ x F
Example A 20 inch wrench is at an angle of 30 degrees with the ground. A force of 40 pounds that makes and angle of 45 degrees with the wrench turns the wrench. Find the torque.
Solution We can write the wrench as the vector 20 cos 30 i + 20 sin 30 j = 17.3 i + 10 j
and the force as -40 cos 75 i - 40 sin 75 j = -10.3 i - 38.6 j
hence, the torque is the magnitude of their cross product:
= -564 inch pounds
To find the volume of the parallelepiped spanned by three vectors u, v
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, and w, we find the triple product:
Volume = u. (v x w)
This can be found by computing the determinate of the three vectors:
Example
Find the volume of the parallelepiped spanned by the vectors
