The dot Product DefinitionWe define the dot product of 2 vectorsv = ai + bj and also w = ci + dj to be v .

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w = ac + bd Notice that the period product of two vectors is a number and not a vector. Because that 3 dimensional vectors, we specify the period product similarly:
Dot Product in R3If v = ai + bj + ck and w = di + ej +fkthen v .w = advertisement + it is in + cf
Examples:If v= 2i + 4j and w= ns + 5jthen v . W= (2)(1) + (4)(5) = 22ExerciseFind the dot product that 2i + j - k andi + 2j The Angle between Two VectorsWe specify the angle theta in between two vectors v and also w by the formulav. W cos q=||v|| ||w|| so the
v . W = ||v|| ||w|| cos q
Two vectors are referred to as orthogonal if your angle is a best angle. Wesee the angles space orthogonal if and only if v . W= 0Example To discover the angle between v = 2i + 3j + k and w= 4i + j + 2k us compute:

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and
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and v . W= 8 + 3 + 2 = 13Hence
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Direction Angles
definition of Direction Cosines permit v = ai + bj + ck be a vector, climate we specify the direction cosines to be the following: a cos a = ||v|| b cos b = ||v|| c cos g = ||v||
Projections and ComponentsSuppose the a car is quit on a steep hill, and let g
it is in the pressure ofgravity exhilaration on it. We can break-up the vector g right into the componentthat is advertise the vehicle down the road and also the component the is advertise thecar ~ above the road. We define
DefinitionLet u and v be a vectors. Climate u have the right to be damaged up into two components, r and also s suchthat r is parallel to v and also s is perpendicular come v. R is called the projection of u ~ above v and also s is called the component of u perpendicular to v.
We view that ||u|| ||v|| ||projvu || u .
v= ||u|| ||v|| cos q = ||u|| = ||v|| ||projvu ||
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hence
u . V ||projvu || = ||v||
We have the right to calculate the projection of u top top v through the formula:
u . Vprojvu =v||v||2
Notice that this works since if we take magnitudes that both political parties we obtain thatu . V
| |projvu|| =||v|| ||v||2and the right hand side simplifies come the formula above. The direction is correct since the appropriate hand next of the formula is a consistent multiple ofv for this reason the estimate vector is in the direction of v together required.To discover the vector s, notification from the diagram thatprojvu + s =uso thats =u - projv uWorkThe work-related done through a constant force F follow me PQ is provided by
W = F . PQ
ExampleFind the work-related done versus gravity to move a 10 kgbaby from the suggest (2,3) come the point (5,7)?SolutionWe have that the pressure vector is F= ma = (10)(-9.8j) = -98jand the displacement vector is v= (5 - 2) i + (7 - 3) j = 3i+ 4jThe work-related is the period productW = F .v = (-98j) .(3i + 4j)= (0)(3) + (-98)(4) = -392Notice the negative sign verifies that the work-related is done versus gravity.Hence, that takes 392 J of work to move the baby.TorqueSuppose you room skiing and also have a disastrous fall. Your body spins aroundand girlfriend ski stays in place (do not shot this at home). With proper bindings her bindings willrelease and also your ski will certainly come off. The bindings identify that a forcehas been applied. This force is dubbed torque. To compute itwe use the cross create of 2 vectors which not only offers the torque,but additionally produces the direction the is perpendicular to both the pressure andthe direction of the leg.

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The overcome Product between Two Vectors
meaning Let u = ai + bj + ck andv = di + ej + fk be vectors then we specify the cross product v x w by the determinant of the matrix:
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We deserve to compute this determinant together
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=
(bf - ce)i + (cd - af) j +(ae - bd) kExampleFind the overcome product u x v ifu = 2i + j - 3kv = 4j + 5kSolutionWe calculate
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= 17i - 10j + 8kIf girlfriend need more help watch the lecture notes for mathematics 103 B on matrices.ExercisesFind u x v when u = 3i + j - 2k, v = ns - k u = 2i - 4j - k, v = 3i- j + 2k notification that because switching the order of 2 rows of a determinant changesthe sign of the determinant, we have actually u x v =-v x uGeometry and also the cross ProductLet u and v be vectors and consider the parallelogram thatthe 2 vectors make. Then
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||u x v|| = Area the the Parallelogramand the direction the u x v is a best angle come the parallelogramthat follows the right hand ruleNote: For i x j the size is 1 and also the direction isk, thus i x j = k.ExerciseFind j x k and i x kTorque RevisitedWe specify the speak (or the moment M of a force F about a suggest Q) together
M = PQ x F
ExampleA 20 inch wrench is in ~ an angle of 30 degrees withthe ground. A force of 40 pounds that makes and angle that 45degreeswith the wrench transforms the wrench. Find the torque.Solution We deserve to write the wrench together the vector 20 cos 30 ns +20 sin 30 j = 17.3 ns + 10 jand the pressure as -40 cos 75 i
- 40 sin 75 j = -10.3i - 38.6 jhence, the talk is the magnitude of their cross product:
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= -564 inch poundsParallelepipedsTo find the volume of the parallelepiped extended by three vectors u
,v, and also w, we find the triple product:
Volume = u . (v x w)
This can be found by computing the determinate the the three vectors:
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Example
Findthe volume of the parallelepiped spanned by the vectorsu = v= w =SolutionWefind
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