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.