vectors maggie ambrose maddy farber. hook… component form of a vector if v is a vector in a plane...
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Vectors
Maggie Ambrose
Maddy Farber
Hook…
Component Form of a Vector
If v is a vector in a plane whose initial point is the origin and whose terminal point is , then the component form of is given by .
The coordinates of and are called the components of .
1v
1 2,v v
2vv
v 1 2,v v v
Magnitude of a Vector
The magnitude is the length of a vector. Let In a 3D coordinate plane, the length is
found in the same way. Let
1 2,v v v2 21 2v v v
1 2 3, ,v v v v2 2 21 2 3v v v v
Find the component and length of the vector v that has initial point (3,-7) and terminal point (-2,5).
Scalar Multiple of a Vector
Let and let be a scalar. The scalar multiple of and is the
vector . The magnitude of the scalar multiple is
equal to the scalar times the magnitude of .
1 2,u u u kk u
1 2,k ku kuu
u
Find the scalar multiple. Let k=6 and let u=2i-j.
Unit Vector
If , then is a unit vector. If is a nonzero vector in the plane, then
the vector
has a magnitude of 1 in the same direction as .
In a 3D coordinate plane, the unit vector is found the same way.
1v vv
vu
v
v
Find a unit vector in the direction of v=-2i+5j.
Dot Product
The dot product of and
is The dot product and
is The dot product of u and v can also be
written as
1 2,u u u
1 2,v v v 1 1 2 2u v u v u v
1 2 3, ,u u u u
1 2 3, ,v v v v 1 1 2 2 3 3u v u v u v u v
cos( )u v u v
Given u=2i-2j and v=5i+8j, find the dot product of u and v.
Angle Between Two Vectors
The angle between two nonzero vectors is the angle , , between their respective standard position vectors.
If theta is the angle between two nonzero vectors u and v, then
0
cos( )u v
u v
For u=3i-j+2k and v=-4i+2k, find the angle between u and v.
Orthogonal vs. Parallel
Orthogonal vectors are perpendicular. The vectors and are orthogonal if
, or if the angle between them is The vectors and are parallel if they
are scalar multiples of each other, or the angle between them is zero.
u v0u v
2
u v
Given u=j+6k and v=i-2j-k, determine whether u and v are orthogonal, parallel, or neither.
Projection
If and are nonzero vectors, then the projection of onto is given by
2v
u vproj u v
v
uu
vv
u
v
projection of u onto v
Find the projection of onto . Let and 3 5 2u i j k 7 2v i j k
u v
Bibliography
Larson, Roland E., Robert P. Hostetler, and Bruce H. Edwards. Calculus. 5th ed. Washington, D.C.: D.C. Heath and Company, 1994.