Study Guide for Prelim 2

Study Guide for Prelim 2 - 1 Study Guide for Prelim 2 Math...

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1 Study Guide for Prelim 2 Math 192 - Spring 1997 Written by Don Allers, Revised by Sean Carver 1 Vectors Idea: A vector v is defined as a directed line segment. Two vectors are equal if they have the same length and the same direction. Vectors in the Plane Idea: Any vector in the plane can be represented as v = a i + b j for some constants a and b . The Vector Connecting 2 Points Idea: Think of the line segment connecting two points as a vector. Method: The vector from the point P = ( x 1 , y 1 ) to the point Q = ( x 2 , y 2 ) is defined to be --→ PQ = v = ( x 2 - x 1 ) i + ( y 2 - y 1 ) j . Eg: What is the vector from the first given point to the second? (a) P = (3 , - 2) , Q = ( - 3 , 2) , (b) R = (1 , 1) , S = (0 , 2 . 6) . Addition and Subtraction of Vectors Idea: We can add two vectors geometrically by drawing them head to tail (so that the tail of the second coincides with the head of the first). The sum then connects the tail of the first to the head of the second. We can subtract two vectors by drawing them so that their tails coincide. The vector v - w will then connect the head of w to the head of v . Note that v - w = v + ( - w ). Method: If v = a i + b j and w = c i + d j , then v + w = ( a + c ) i + ( b + d ) j v - w = ( a - c ) i + ( b - d ) j . Eg: Compute the following: (a) (6 i - 3 j ) + ( - 6 i + 3 j ) , (b) ( i + j ) + i , (c) --→ PQ + -→ RS from above . Scalar Multiplication Idea: For a constant c > 0, the vector c v represents the vector with the same direction as v , but length equal to c times the length of v . If c < 0, c v represents the vector with direction opposite to v and with length | c | times the length of v . Method: For v = a i + b j , we have c v = ( ca ) i + ( cb ) j . Eg: Compute: (a) π ( i + j ) , (b) 1 3 (2 i + 3 j ) , (c) 5 ~ PQ + 1 2 ~ RS. Magnitude and Direction Idea: Since v represents a directed line segment, it has a length and a direction. Notation: The length (magnitude) of the vector v = a i + b j is denoted | v | and equals a 2 + b 2 . Def. A unit vector is a vector whose length equals 1. Def. The direction of a non-zero vector v is defined to be the unit vector with the same direction as v . The zero vector has no direction. Method: Given any non-zero vector v , the vector 1 | v | v is a scalar multiple of v so it has the same direction. Furthermore, its length is 1, so v | v | is the direction of v , as defined above. We can write v = | v | v | v | to express v as the product of its length and direction.
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2 1 VECTORS Eg: What is the length of: (a) 3 i - 2 j , (b) 3 i + 6 j . Find a unit vector in the opposite direction of: (c) π i - π j , (d) 3 i + 12 j . Vectors In Space Idea: A vector A in space can be represented as A = a 1 i + a 2 j + a 3 k for some constants a 1 , a 2 , and a 3 . All of the definitions above can be extended to apply in this context. The Vector Connecting 2 Points Method: The vector between points P = ( x 1 , y 1 , z 1 ) and Q = ( x 2 , y 2 , z 2 ) is defined by --→ PQ = A = ( x 2 - x 1 ) i + ( y 2 - y 1 ) j + ( z 2 - z 1 ) k . Eg: What is the vector connecting the first given point to the second?
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