8.2.3 Vectors (continued)
+ = 7 7 49 2, 2 2
The magnitude of vector CD is
and the magnitude of vector GH is + = 3 3 9 2. 2 2 These two vectors are not equal because they have different magnitudes, but they are parallel because they have the same direction.
Example 5 Real-World Application
The magnitude and direction of a vector that is a sum of two vectors is determined in this application example. The magnitude and direction of the two component vectors are given. Begin by finding the component form of the two vectors. The kayaker’s vector components can be determined from the length of the hypotenuse and the angle made with the x -axis. The x component is 5 cos 40 ° ≈ 3.8 mi/h. The y component is 5 sin 40 ° ≈ 3.2 mi/h. The kayaker’s vector is 〈 3.8, 3.2 〉 . The x component of the current vector is 1, and its y component is 0. The current’s vector is 〈 1, 0 〉 . 〈 3.8, 3.2 〉 + 〈 1, 0 〉 = 〈 3.8 + 1, 3.2 + 0 〉 = 〈 4.8, 3.2 〉 . The actual speed of the kayak (including both the paddling of the kayak and the current) can be obtained using the Distance Formula. Since the initial point of the resultant vector is the origin, the Distance Formula is simpler in form. The approximate magnitude of the resultant vector is 5.8 mi/h. The actual direction of the kayaker is the measure of the angle formed by the resultant vector and the x -axis. The measure of the angle is the inverse tangent of 3.2/4.8, or approximately 34 ° . This is equivalent to the direction N 56 ° E, the angle formed by the resultant vector and the y -axis. To combine the vectors, sum their components. The resultant vector is
92
Made with FlippingBook - PDF hosting