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

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Vectors A vector is 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.

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

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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. 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 We write The next slide gives a general definition of vector addition: The next slide gives a general definition of vector addition:

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

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Vector Addition (cont’d) We can instead draw another copy of v with the same initial point as u. 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. Completing the parallelogram as on the next slide, we see that u + v = v + u. This gives another way to form the sum: 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. 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:

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

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

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

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Example If a and b are the vectors shown on the left, draw a – 2b. 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. 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): We place its tail at the tip of a and then use the Triangle Law to draw a + (– 2b):

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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 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 (a 1, a 2 ) or (a 1, a 2 ) or (a 1, a 2, a 3 ), (a 1, a 2, a 3 ), depending on whether our coordinate system is two- or three-dimensional.

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

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Representations (cont’d) In three dimensions, is the position vector of the point P(a 1, a 2, a 3 ),. In three dimensions, is the position vector of the point P(a 1, a 2, a 3 ),. Vector addition leads to the following result: Vector addition leads to the following result:

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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 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: The distance formula gives:

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Using Components The next slide illustrates the following rules: The next slide illustrates the following rules:

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Using Components (cont’d)

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Example If the vectors a + b, a – b, 3b, and 2a + 5b. If the vectors a + b, a – b, 3b, and 2a + 5b. Solution Solution

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The Set V n We denote by… We denote by… V 2 the set of all two-dimensional vectors and V 2 the set of all two-dimensional vectors and V 3 the set of all three-dimensional vectors. V 3 the set of all three-dimensional vectors. We will later need to consider the set V n of all n-dimensional vectors. We will later need to consider the set V n of all n-dimensional vectors. An n-dimensional vector is an ordered n- tuple An n-dimensional vector is an ordered n- tuple

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

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Standard Basis Vectors Three vectors in V 3 play a special role: Three vectors in V 3 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. 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 In two dimensions we put

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Standard Basis Vectors (cont’d)

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If If Thus, any vector in V 3 can be expressed in terms of i, j, and k. Thus, any vector in V 3 can be expressed in terms of i, j, and k.

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

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Unit Vectors A unit vector is a vector whose length is 1. A unit vector is a vector whose length is 1. For instance, i, j, and k are all unit vectors. For instance, i, j, and k are all unit vectors. If a is not the zero vector 0, then the unit vector that has the same direction as a is If a is not the zero vector 0, then the unit vector that has the same direction as a is

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

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Review Vectors Vectors Combining vectors Combining vectors Addition Addition Scalar multiplication Scalar multiplication Components Components Properties of vectors Properties of vectors

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