8.2.3 Vectors (continued) Example 2 Finding the Magnitude of a Vector
A vector in component form is drawn on the coordinate plane and its magnitude is determined in this example. When the origin (0, 0) is used as the initial point, the terminal point has the coordinates of the component values ( − 2, 5). Use the Distance Formula to calculate the magnitude of the vector. In this case, | 〈− 2, 5 〉 | ≈ 5.4.
Example 3 Finding the Direction of a Vector
A vector in component form is drawn on the coordinate plane and its direction is determined in this example. When the origin (0, 0) is used as the initial point, the terminal point has the coordinates of the component values (3, 5). The tangent of the angle formed with the x -axis is the y component over the x component. The measure of the angle is tan − 1 (5/3) ≈ 59 ° . The direction of the vector is approximately 59 ° .
Example 4 Identifying Equal and Parallel Vectors
Equal and parallel vectors are identified in this example. To compare the vectors write them in component form. Vector AB = 〈 2, 5 〉 , vector CD = 〈 7, 7 〉 , vector EF = 〈 2, 5 〉 , and vector GH = 〈 3, 3 〉 . Equal vectors have the same magnitude and direction. Vectors with the same direction have the same angle with the horizontal axis. The angles of these vectors are equal to tan − 1 ( y / x ). Vectors AB and EF have the same direction and vectors CD and GH have the same direction. The magnitudes of the vectors can be determined using the Distance Formula. |AB| = |EF| = = = + = AB EF | | | | 2 5 29. 2 2 Vectors AB and EF are equal, because they have the same magnitude and direction.
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