Q:

# How do you add two vectors that are not perpendicular or parallel?

A:

For each vector, the angle of the vector to the horizontal must be determined. Using this angle, the vectors can be split into their horizontal and vertical components using the trigonometric functions sine and cosine. The horizontal components for the vectors are solved separately from the vertical. The combined horizontal and vertical components are solved using the Pythagorean theorem to reach the final answer.

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The angle of each vector should be measured to the horizontal X-axis. Using the angle obtained, the length of the vector can be multiplied by the cosine of the angle to calculate the horizontal component of the vector. The length of the vector should also be multiplied by the sine of the angle to calculate the vertical component of the vector. Once the horizontal and vertical components of the vector are calculated, the vector at an angle is no longer required for the vector addition calculation. This step is repeated for all the vectors given in the problem. Once all the vectors are expressed in terms of only their horizontal and vertical components, the total horizontal component for all vectors and the total vertical component is calculated using simple vector addition. If both the horizontal components, or vertical components, are in the same direction, they are added; if in opposite direction, they are subtracted. This leaves one final horizontal component and one vertical component. Since these are at right angles to each other and the lengths are known, the Pythagorean theorem is used to calculate the magnitude of the resultant vector. The angle of the vector to the horizontal gives the direction of the vector.

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