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Section 9.2 Vectors

Section 9.2 Vectors. Goals Introduce vectors . Begin to discuss operations with vectors and vector components . Give properties of vectors. Vectors. A vector is a quantity (such as displacement or velocity or force) that has both magnitude and direction.

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Section 9.2 Vectors

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  1. Section 9.2Vectors • Goals • Introduce vectors. • Begin to discuss operations with vectors and vector components. • Give properties of vectors.

  2. Vectors • A vectoris a quantity (such as displacement or velocity or force) that has both magnitude and direction. • A vector is often represented by an arrow or a directed line segment. • The length of the arrow represents the magnitude of the vector and • the arrow points in the direction of the vector.

  3. Vectors (cont’d) • For instance, if a particle moves along a line segment from point A to point B, then the corresponding displacement vectorv has initial pointA (the tail) and terminal point B (the tip). • We write • Note

  4. Vector Addition • If a particle moves from A to B, and then from B to C, the net effect is that the particle moves from A to C. • We write • The next slide givesa general definition ofvector addition:

  5. Vector Addition (cont’d) • The figure shows why this definition is sometimes called the Triangle Law.

  6. Vector Addition (cont’d) • We can instead draw another copy of v with the same initial point as u. • Completing the parallelogram as on the next slide, we see that u + v = v + u. • This gives another way to form the sum: • If we place u and v so they start at the same point, then u + v lies along the diagonal of the parallelogram with u and v as sides. This is called the Parallelogram Law:

  7. Scalar Multiplication • The following definition shows how we multiply a vector by a real number c:

  8. Scalar Multiplication (cont’d) • Note that… • Two nonzero vectors are parallel if they are scalar multiples of one another. • In particular, the vector –v = (–1)v (called the negative of v) • has the same length as v but • points in the opposite direction.

  9. Vector Subtraction • By the difference of two vectors we mean u – v = u + (–v) • We can construct u – v … • by first drawing –v and then adding it to u by the Parallelogram Law, or • by means of the Triangle Law.

  10. Example • If a and b are the vectors shown on the left , draw a – 2b. • Solution We first draw the vector –2b pointing in the direction opposite to b and twice as long. • We place its tail at the tip of a and then use the Triangle Law to draw a + (– 2b):

  11. Components • If we place the initial point of a vector a at the origin of a rectangular coordinate system, then the terminal point of a has coordinates of the form • (a1, a2) or • (a1, a2, a3), depending on whether our coordinate system is two- or three-dimensional.

  12. Representations • The vectors shown are all equivalent to the vector terminal point is P(3, 2). • We can think of all these geometric vectors as representations of the vector • The particular representationorigin to the point P(3, 2) is called the position vector of the point P :

  13. Representations (cont’d) • In three dimensions,is the position vector of the point P(a1, a2, a3),. • Vector addition leads to the following result:

  14. Magnitude • The magnitude or length of the vector v is the length of any of its representations and is denoted by the symbol |v|. • The distance formula gives:

  15. Using Components • The next slide illustrates the following rules:

  16. Using Components (cont’d)

  17. Example • Ifthe vectors a + b, a – b, 3b, and 2a + 5b. • Solution

  18. The Set Vn • We denote by… • V2 the set of all two-dimensional vectors and • V3 the set of all three-dimensional vectors. • We will later need to consider the set Vnof all n-dimensional vectors. • An n-dimensional vector is an ordered n-tuple

  19. Properties of Vectors • These properties can be verified either geometrically or algebraically.

  20. Standard Basis Vectors • Three vectors in V3 play a special role: • These vectors have length 1 and point in the directions of the positive x-, y-, and z-axes, as shown on the next slide. • In two dimensions we put

  21. Standard Basis Vectors (cont’d)

  22. Standard Basis Vectors (cont’d) • If • Thus, any vector in V3 can be expressed in terms of i, j, and k.

  23. Example • If a = i + 2j – 3k and b = 4i + 7k, express the vector 2a + 3b in terms of i, j, and k. • Solution

  24. Unit Vectors • A unit vector is a vector whose length is 1. • For instance, i, j, and k are all unit vectors. • If a is not the zero vector0, then the unit vector that has the same direction as a is

  25. Example • Find the unit vector in the direction of2i – j – 2k. • Solution The given vector has length • Thus the unit vector with the same direction is ⅓ (2i – j – 2k) = ⅔i - ⅓j - ⅔k.

  26. Review • Vectors • Combining vectors • Addition • Scalar multiplication • Components • Properties of vectors

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