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If you were to hit a golf ball you would be able to find the distance that it traveled horizontally and vertically if you know the original vectors.

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Knowing the angle could help when programming something to move

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Spherical trig? Use the angle between the vectors to help find the arc distance? Location?

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when throwing/launching an object, the angle that it launches from will determine the distance it travels

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If you have two forces pushing on an object (bulldozers on rocks) you can find the angle at which the rocks will move.

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If someone bowls, they can use the angle between two vectors to find the angle at which the ball should be bowled at, and where it will reach.

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The definition of vectors contains   "direction" in its definition, and in the math world, angles between vectors completes the definition.
Example: A line on google map that points me towards sonic for a "SuperSonic double cheeseburger".

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finding the angle between two vector points can allow

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if you are going swimming there can riptides

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lattice in musical tuning

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Finding Vector angles can help you find specific directions to a certain place. Example: What direction do I have to go to to find the nearest Wendy's?

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Specific directions or information between two points.

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Being able to find vectors are important with many jobs. For instance meteorology, aviation, manufacturers, or wave patterns

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Finding the angle can help w direction

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Say a ball is moving forward with a certain force and you push the ball upward. The direction that the ball will travel will be the sum of the 2 vectors.

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It can be used in baseball and basketball for finding the trajectory of the ball

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Finding the angle between two vectors can help in Physics, specifically kinematics.Kinematics is the basic motion of objects, such as a ball being thrown into the air and it returning to the ground. A projectile launched into the air will vary based on the angle it is being launched at, the vector components along the y axis and the x axis will determine the angle and the distance it will travel