38 lines
1.9 KiB
Markdown
38 lines
1.9 KiB
Markdown
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---
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title: Addition and Scalar Multiplication
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---
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## Addition and Scalar Multiplication
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When working with vectors, the two most common operations are addition of vectors and multiplication by a scalar.
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### Vector Addition
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Vector addition can be visualized as follows:
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1. Take the "tail" (The end without an arrow/the origin of the vector) of the second vector, and connect it (unaltered) to the "tip" (The pointed/arrow end) of the first vector. Now, if you create a new vector from the tail of the first vector to the tip of the second vector, you will be left with the sum of the two vectors!
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2. Subtracting two vectors is almost the same. However, you must flip the direction of the second vector, and then proceed with connecting it to the first.
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Obviously, you don't want to have to draw and connect vectors every single time you want to do vector addition. Luckily, the solution is much simpler in practice.
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Assuming you have two vectors <1,2> and <5,-4>, all you have to do is add the corresponding components:
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<pre> <1,2> + <5,-4> = <1 + 5, 2 + (-4)> = <6, -2> </pre>
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This works with vectors of as many dimensions as you want, as long as the sizes of the two vectors added are the same. For example, adding <4, 4, -5, 0> and <2, 4, -1, -29>:
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<pre> <4, 4, -5, 0> + <2, 4, -1, -29> = <4 + 2, 4 + 4, -5 + (-1), 0 + (-29)> = <6, 8, -6, -29> </pre>
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### Scalar Multiplication
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When multiplying a vector by a scalar, you can think of it as increasing its magnitude.
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For example, when multiplying vector <2, 3> by 2:
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<pre> 2 * <2,3> = <2 * 2, 2 * 3> = <4, 6> </pre>
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The direction is preserved - just the magnitude is increased by a factor of 2.
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However, when we multiply by a negative number, the direction is reversed. When multiplying vector <2, 3> by -2:
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<pre> -2 * <2, 3> = <-2 * 2, -2 * 3> = <-4, -6> </pre>
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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