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.