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Vectors

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 .

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Vectors

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  1. Vectors Maggie Ambrose Maddy Farber

  2. Hook…

  3. 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 .

  4. 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

  5. Find the component and length of the vector v that has initial point (3,-7) and terminal point (-2,5).

  6. 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 .

  7. Find the scalar multiple. Let k=6 and let u=2i-j.

  8. 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.

  9. Find a unit vector in the direction of v=-2i+5j.

  10. 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

  11. Given u=2i-2j and v=5i+8j, findthe dot product of u and v.

  12. 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

  13. For u=3i-j+2k and v=-4i+2k, find the angle between u and v.

  14. 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.

  15. Given u=j+6k and v=i-2j-k, determine whether u and v are orthogonal, parallel, or neither.

  16. Projection • If and are nonzero vectors, then the projection of onto is given by u v projection of u onto v

  17. Find the projection of onto . Let and

  18. 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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