mcv4ua_unit_3_lesson_15.pdf - MCV4U-A 15 Properties of...

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15 MCV4U-A Properties of Vectors and Scalar Multiplication
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www.ilc.org Copyright © 2008 The Ontario Educational Communications Authority. All rights reserved. Calculus and Vectors MCV4U-A Lesson 15, page 1 Introduction In Lesson 14, you learned about vectors in three-space, and adding and subtracting vectors. In this lesson, you will learn about scalar multiplication, and some of the properties or rules for manipulating vectors. You will also use this knowledge to answer some real-world problems involving vectors. What type of real-world problems? As an example, you can use vectors to determine which direction a pilot should steer an airplane if you know the distance to the destination airport as well as the direction and velocity of the wind and of the airplane. Estimated Hours for Completing This Lesson Multiplying a Vector by a Scalar 0.5 Properties of Vector Arithmetic 1 Solving Real-World Problems Using Vectors 2 Key Questions 1.5 What You Will Learn After completing this lesson, you will be able to add and subtract scalar multiples of vectors in two-space and three-space apply properties of vector arithmetic such as the associative law and distributive law solve real-world problems involving vector addition, vector subtraction, and scalar multiplication
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Lesson 15, page 2 Calculus and Vectors MCV4U-A Copyright © 2008 The Ontario Educational Communications Authority. All rights reserved. www.ilc.org Multiplying a Vector by a Scalar In earlier lessons, you’ve seen that vectors can be represented as directed line segments (geometric vectors), and they can be represented in Cartesian form using coordinates. You have learned how to add and subtract vectors in both of those forms. Not only can you add two vectors or subtract two vectors, it’s also possible to multiply a vector by a scalar. If v is any vector and k is any scalar, then kv is a new vector with a length that is k times the length of v . Vectors that are scalar multiples of v are described as parallel to v , although the magnitude may change depending on the value of k . If k is negative, then kv will be parallel to v , but it will face in the opposite direction to v . Consider some examples, using both geometric vectors and Cartesian vectors. Scalar Multiplication of Geometric Vectors If v is a vector represented by a directed line segment, then kv is a vector parallel to v , whose magnitude and direction has been modified appropriately. If v is a vector quantity such as a velocity or a force, it is physically meaningful to talk about multiplying v by a positive or negative number. Examples A hot-air balloon is travelling N10°E at a speed of 30 km/h. Let v be the vector representing the balloon’s velocity. Find the magnitude and direction of each of the following vectors: a) 2 v b) 0.8 v c) v d) 1 2 v
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www.ilc.org Copyright © 2008 The Ontario Educational Communications Authority. All rights reserved.
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