# 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.

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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.

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 

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.

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.

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

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

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.

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 u v projection of u onto v

Find the projection of onto. Let and

Bibliography Larson, Roland E., Robert P. Hostetler, and Bruce H. Edwards. Calculus. 5th ed. Washington, D.C.: D.C. Heath and Company, 1994.

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